ALSOBYSTEPHENHAWKING
ABrieferHistoryofTime
BlackHolesandBabyUniversesandOtherEssays
TheIllustratedABriefHistoryofTime
TheUniverseinaNutshell
TheGrandDesign
FORCHILDREN
George’sSecretKeytotheUniverse(withLucyHawking)
George’sCosmicTreasureHunt(withLucyHawking)
ABRIEFHISTORYOFTIME
ABantamBook
PublishingHistory
BantamillustratedhardcovereditionpublishedNovember1996
Bantamhardcoveredition/September1998
Bantamtradepaperbackedition/September1998
Allrightsreserved.
Copyright©1988,1996byStephenHawking
Illustrationscopyright©1988byRonMiller
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Hawking,S.W.(StephenW.)
Abriefhistoryoftime/StephenHawking.
p.cm.
Includesindex.
eISBN:978-0-55389692-3
1.Cosmology.I.Title.
QB981.H3771998
523.1—dc2198-21874
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v3.1
CONTENTS
Cover
OtherBooksbyThisAuthor
TitlePage
Copyright
FOREWORD
ChapterOneOurPictureoftheUniverse
ChapterTwoSpaceandTime
ChapterThreeTheExpandingUniverse
ChapterFourTheUncertaintyPrinciple
ChapterFiveElementaryParticlesandtheForcesofNature
ChapterSixBlackHoles
ChapterSevenBlackHolesAin’tSoBlack
ChapterEightTheOriginandFateoftheUniverse
ChapterNineTheArrowofTime
ChapterTenWormholesandTimeTravel
ChapterElevenTheUnificationofPhysics
ChapterTwelveConclusion
ALBERTEINSTEIN
GALILEOGALILEI
ISAACNEWTON
GLOSSARY
ACKNOWLEDGMENTS
AbouttheAuthor
I
FOREWORD
didn’t write a foreword to the original edition of A Brief History of
Time.ThatwasdonebyCarlSagan.Instead,Iwroteashortpiecetitled
“Acknowledgments”inwhichIwasadvisedtothankeveryone.Someof
thefoundationsthathadgivenmesupportweren’ttoopleasedtohave
been mentioned, however, because it led to a great increase in
applications.
Idon’tthinkanyone,mypublishers,myagent,ormyself,expectedthe
booktodoanythinglikeaswellasitdid.ItwasintheLondonSunday
Times best-seller list for 237 weeks, longer than any other book
(apparently, the Bible and Shakespeare aren’t counted). It has been
translated into something like forty languages and has sold about one
copyforevery750men,women,andchildrenintheworld.AsNathan
MyhrvoldofMicrosoft(aformerpost-docofmine)remarked:Ihavesold
morebooksonphysicsthanMadonnahasonsex.
The success of A Brief History indicates that there is widespread
interestinthebigquestionslike:Wheredidwecomefrom?Andwhyis
theuniversethewayitis?
I have taken the opportunity to update the book and include new
theoretical and observational results obtained since the book was first
published(onAprilFools’Day,1988).Ihaveincludedanewchapteron
wormholesandtimetravel.Einstein’sGeneralTheoryofRelativityseems
to offer the possibility that we could create and maintain wormholes,
littletubesthatconnectdifferentregionsofspace-time.Ifso,wemight
beabletousethemforrapidtravelaroundthegalaxyortravelbackin
time.Ofcourse,wehavenotseenanyonefromthefuture(orhavewe?)
butIdiscussapossibleexplanationforthis.
I also describe the progress that has been made recently in finding
“dualities”orcorrespondencesbetweenapparentlydifferenttheoriesof
physics. These correspondences are a strong indication that there is a
completeunifiedtheoryofphysics,buttheyalsosuggestthatitmaynot
bepossibletoexpressthistheory ina singlefundamental formulation.
Instead, we may have to use different reflections of the underlying
theory in different situations. It might be like our being unable to
represent the surface of the earth on a single map and having to use
different maps in different regions. This would be a revolution in our
viewoftheunificationofthelawsofsciencebutitwouldnotchangethe
mostimportantpoint:thattheuniverseisgovernedbyasetofrational
lawsthatwecandiscoverandunderstand.
Ontheobservationalside,byfarthemostimportantdevelopmenthas
been the measurement of fluctuations in the cosmic microwave
background radiation by COBE (the Cosmic Background Explorer
satellite)andothercollaborations.Thesefluctuationsarethefingerprints
of creation, tiny initial irregularities in the otherwise smooth and
uniform early universe that later grew into galaxies, stars, and all the
structureswesee aroundus.Their formagreeswith thepredictionsof
the proposal that the universe has no boundaries or edges in the
imaginarytime direction;butfurther observationswill be necessaryto
distinguish this proposal from other possible explanations for the
fluctuationsinthebackground.However,withinafewyearsweshould
know whether we can believe that we live in a universe that is
completelyself-containedandwithoutbeginningorend.
StephenHawking
A
CHAPTER1
OURPICTUREOF
THEUNIVERSE
well-knownscientist(somesayitwasBertrandRussell)oncegavea
public lecture on astronomy. He described how the earth orbits
aroundthesunandhowthesun,inturn,orbitsaroundthecenterofa
vast collection of stars called our galaxy. At the end of the lecture, a
littleoldladyatthebackoftheroomgotupandsaid:“Whatyouhave
toldusisrubbish.Theworldisreallyaflatplatesupportedontheback
ofagianttortoise.”Thescientistgaveasuperiorsmilebeforereplying,
“Whatisthetortoisestandingon?”“You’reveryclever,youngman,very
clever,”saidtheoldlady.“Butit’sturtlesallthewaydown!”
Mostpeoplewouldfindthepictureofouruniverseasaninfinitetower
of tortoises rather ridiculous, but why do we think we know better?
Whatdoweknowabouttheuniverse,andhowdoweknowit?Where
did the universe come from, and where is it going? Did the universe
have a beginning, and if so, what happened before then? What is the
natureoftime?Willitevercometoanend?Canwegobackintime?
Recentbreakthroughsinphysics,madepossibleinpartbyfantasticnew
technologies,suggest answers to some of these longstanding questions.
Somedaytheseanswersmayseemasobvioustousastheearthorbiting
the sun—or perhaps as ridiculous as a tower of tortoises. Only time
(whateverthatmaybe)willtell.
Aslongagoas340B.C.theGreekphilosopherAristotle,inhisbookOn
theHeavens,wasabletoputforwardtwogoodargumentsforbelieving
that the earth was a round sphere rather than a flat plate. First, he
realized that eclipses of the moon were caused by the earth coming
between the sun and the moon. The earth’s shadow on the moon was
alwaysround,whichwouldbetrueonlyiftheearthwasspherical.Ifthe
earthhadbeenaflatdisk,theshadowwouldhavebeenelongatedand
elliptical,unlesstheeclipsealwaysoccurredatatimewhenthesunwas
directlyunderthecenterofthedisk.Second,theGreeksknewfromtheir
travelsthattheNorthStarappearedlowerintheskywhenviewedinthe
south than it did in more northerly regions. (Since the North Star lies
overtheNorthPole,itappearstobedirectlyaboveanobserveratthe
NorthPole,buttosomeonelookingfromtheequator,itappearstolie
justatthehorizon.Fromthedifferenceintheapparentpositionofthe
NorthStarinEgyptandGreece,Aristotleevenquotedanestimatethat
the distance around the earth was 400,000 stadia. It is not known
exactly what length a stadium was, but it may have been about 200
yards,whichwouldmakeAristotle’sestimateabouttwicethecurrently
accepted figure. The Greeks even had a third argument that the earth
mustberound,forwhyelsedoesonefirstseethesailsofashipcoming
overthehorizon,andonlylaterseethehull?
Aristotlethoughttheearthwasstationaryandthatthesun,themoon,
theplanets,and thestars movedin circularorbits aboutthe earth.He
believedthisbecausehefelt,formysticalreasons,thattheearthwasthe
center of the universe, and that circular motion was the most perfect.
ThisideawaselaboratedbyPtolemyinthesecondcenturyA.D.intoa
completecosmologicalmodel.Theearthstoodatthecenter,surrounded
byeightspheresthatcarriedthemoon,thesun,thestars,andthefive
planets known at the time, Mercury, Venus, Mars, Jupiter, and Saturn
(Fig.1.1).Theplanetsthemselvesmovedonsmallercirclesattachedto
theirrespectivespheresinordertoaccountfortheirrathercomplicated
observed paths in the sky. The outermost sphere carried the so-called
fixed stars, which always stay in the same positions relative to each
otherbutwhichrotatetogetheracrossthesky.Whatlaybeyondthelast
sphere was never made very clear, but it certainly was not part of
mankind’sobservableuniverse.
Ptolemy’smodelprovidedareasonablyaccuratesystemforpredicting
thepositionsofheavenlybodiesinthesky.Butinordertopredictthese
positionscorrectly,Ptolemyhadtomakeanassumptionthatthemoon
followedapaththatsometimesbroughtittwiceasclosetotheearthas
atothertimes.Andthatmeantthatthemoonoughtsometimestoappear
twice as big as at other times! Ptolemy recognized this flaw, but
nevertheless his model was generally, although not universally,
accepted.It was adopted by the Christian church as the picture of the
universe that was in accordance with Scripture, for it had the great
advantagethatitleft lotsofroomoutsidethe sphereoffixed starsfor
heavenandhell.
FIGURE1.1
Asimplermodel,however,wasproposedin1514byaPolishpriest,
Nicholas Copernicus. (At first, perhaps for fear of being branded a
heretic by his church, Copernicus circulated his model anonymously.)
Hisideawasthatthesunwasstationaryatthecenterandthattheearth
and the planets moved in circular orbits around the sun. Nearly a
century passed before this idea was taken seriously. Then two
astronomers—the German, Johannes Kepler, and the Italian, Galileo
Galilei—started publicly to support the Copernican theory, despite the
factthattheorbitsitpredicteddidnotquitematchtheonesobserved.
The death blow to the Aristotelian/Ptolemaic theory came in 1609. In
thatyear,Galileostartedobservingthenightskywithatelescope,which
had just been invented. When he looked at the planet Jupiter, Galileo
foundthatitwasaccompaniedbyseveralsmallsatellitesormoonsthat
orbited around it. This implied that everything did not have to orbit
directlyaroundtheearth,asAristotleandPtolemyhadthought.(Itwas,
of course, still possible to believe that the earth was stationary at the
centeroftheuniverseandthatthemoonsofJupitermovedonextremely
complicated paths around the earth, giving the appearance that they
orbitedJupiter.However,Copernicus’stheorywasmuchsimpler.)Atthe
same time, Johannes Kepler had modified Copernicus’s theory,
suggestingthattheplanetsmovednotincirclesbutinellipses(anellipse
is an elongated circle). The predictions now finally matched the
observations.
AsfarasKeplerwasconcerned,ellipticalorbitsweremerelyanadhoc
hypothesis, and a rather repugnant one at that, because ellipses were
clearly less perfect than circles. Having discovered almost by accident
that elliptical orbits fit the observations well, he could not reconcile
them with his idea that the planets were made to orbit the sun by
magneticforces.Anexplanationwasprovidedonlymuchlater,in1687,
when Sir Isaac Newton published his Philosophiae Naturalis Principia
Mathematica,probablythemostimportantsingleworkeverpublishedin
thephysicalsciences.InitNewtonnotonlyputforwardatheoryofhow
bodiesmoveinspaceandtime,buthealsodevelopedthecomplicated
mathematics needed to analyze those motions. In addition, Newton
postulatedalawofuniversalgravitationaccordingtowhicheachbody
intheuniverse was attracted toward every other body by a forcethat
wasstronger the moremassive the bodiesand the closerthey were to
each other. It was this same force that caused objects to fall to the
ground. (The story that Newton was inspired by an apple hitting his
headis almost certainly apocryphal.All Newton himself eversaidwas
thattheideaofgravitycametohimashesat“inacontemplativemood”
and“wasoccasionedbythefallofanapple.”)Newtonwentontoshow
that, according to his law, gravity causes the moon to move in an
ellipticalorbitaroundtheearthandcausestheearthandtheplanetsto
followellipticalpathsaroundthesun.
TheCopernicanmodelgotridofPtolemy’scelestialspheres,andwith
them, the idea that the universe had a natural boundary. Since “fixed
stars” did not appear to change their positions apart from a rotation
acrosstheskycausedbytheearthspinningonitsaxis,itbecamenatural
tosupposethatthefixedstarswereobjectslikeoursunbutverymuch
fartheraway.
Newton realized that, according to his theory of gravity, the stars
shouldattracteachother,soitseemedtheycouldnotremainessentially
motionless.Wouldtheynotallfalltogetheratsomepoint?Inaletterin
1691 to Richard Bentley, another leading thinker of his day, Newton
arguedthatthiswouldindeedhappeniftherewereonlyafinitenumber
ofstarsdistributedoverafiniteregionofspace.Buthereasonedthatif,
on the other hand, there were an infinite number of stars, distributed
more or less uniformly over infinite space, this would not happen,
becausetherewouldnotbeanycentralpointforthemtofallto.
Thisargumentisaninstanceofthepitfallsthatyoucanencounterin
talking about infinity. In an infinite universe, every point can be
regarded as the center, because every point has an infinite number of
starsoneachsideofit.Thecorrectapproach,itwasrealizedonlymuch
later,istoconsiderthefinitesituation,inwhichthestarsallfallinon
eachother, and then toask how things changeif one adds more stars
roughlyuniformlydistributedoutsidethisregion.AccordingtoNewton’s
law,theextrastarswouldmakenodifferenceatalltotheoriginalones
onaverage,sothestarswouldfallinjustasfast.Wecanaddasmany
starsaswelike,buttheywillstillalwayscollapseinonthemselves.We
now know it is impossible to have an infinite static model of the
universeinwhichgravityisalwaysattractive.
Itisaninterestingreflectiononthegeneralclimateofthoughtbefore
thetwentiethcenturythatnoonehadsuggestedthattheuniversewas
expanding or contracting. It was generally accepted that either the
universehadexistedforeverinanunchangingstate,orthatithadbeen
createdatafinitetimeinthepastmoreorlessasweobserveittoday.In
partthismayhavebeenduetopeople’stendencytobelieveineternal
truths,aswellasthecomforttheyfoundinthethoughtthateventhough
theymaygrowoldanddie,theuniverseiseternalandunchanging.
EventhosewhorealizedthatNewton’stheoryofgravityshowedthat
theuniversecouldnotbestaticdidnotthinktosuggestthatitmightbe
expanding.Instead,theyattemptedtomodifythetheorybymakingthe
gravitational force repulsive at very large distances. This did not
significantlyaffecttheirpredictionsofthemotionsoftheplanets,butit
allowedaninfinitedistributionofstarstoremaininequilibrium—with
the attractive forces between nearby stars balanced by the repulsive
forcesfromthosethatwerefartheraway.However,wenowbelievesuch
anequilibriumwouldbeunstable:ifthestarsin someregiongotonly
slightly nearer each other, the attractive forces between them would
becomestrongeranddominateovertherepulsiveforcessothatthestars
wouldcontinuetofalltowardeachother.Ontheotherhand,ifthestars
got a bit farther away from each other, the repulsive forces would
dominateanddrivethemfartherapart.
Anotherobjectiontoaninfinitestaticuniverseisnormallyascribedto
theGermanphilosopherHeinrichOlbers,whowroteaboutthistheoryin
1823.Infact,variouscontemporariesofNewtonhadraisedtheproblem,
and the Olbers article was not even the first to contain plausible
argumentsagainstit.Itwas,however,thefirsttobewidelynoted.The
difficultyisthatinaninfinitestaticuniversenearlyeverylineofsight
would end on the surface of a star. Thus one would expect that the
whole sky would be as bright as the sun, even at night. Olbers’s
counterargumentwasthatthelightfromdistantstarswouldbedimmed
by absorption by intervening matter. However, if that happened the
interveningmatterwouldeventuallyheatupuntilitglowedasbrightly
asthestars.Theonlywayofavoidingtheconclusionthatthewholeof
thenightskyshouldbeasbrightasthesurfaceofthesunwouldbeto
assumethatthestarshadnotbeenshiningforeverbuthadturnedonat
somefinitetimeinthepast.Inthatcasetheabsorbingmattermightnot
have heated up yet or the light from distant stars might not yet have
reachedus.Andthatbringsustothequestionofwhatcouldhavecaused
thestarstohaveturnedoninthefirstplace.
The beginning of the universe had, of course, been discussed long
before this. According to a number of early cosmologies and the
Jewish/Christian/Muslimtradition,theuniversestartedatafinite,and
not very distant, time in the past. One argument for such a beginning
wasthefeelingthatitwasnecessarytohave“FirstCause”toexplainthe
existence of the universe. (Within the universe, you always explained
oneeventasbeingcausedbysomeearlierevent,buttheexistenceofthe
universe itself could be explained in this way only if it had some
beginning.)AnotherargumentwasputforwardbySt.Augustineinhis
bookTheCityofGod.Hepointedoutthatcivilizationisprogressingand
we remember who performed this deed or developed that technique.
Thusman,andsoalsoperhapstheuniverse,couldnothavebeenaround
all that long. St. Augustine accepted a date of about 5000 B.C. for the
Creation of the universe according to the book of Genesis. (It is
interestingthatthisisnotsofarfromtheendofthelastIceAge,about
10,000B.C., which is when archaeologists tell us that civilization really
began.)
Aristotle, and most of the other Greek philosophers, on the other
hand,didnotliketheideaofacreationbecauseitsmackedtoomuchof
divineintervention.Theybelieved,therefore,that thehumanrace and
theworldaroundithadexisted,andwouldexist,forever.Theancients
had already considered the argument about progress described above,
andanswereditbysayingthattherehadbeenperiodicfloodsorother
disastersthatrepeatedlysetthehumanracerightbacktothebeginning
ofcivilization.
The questions of whether the universe had a beginning in time and
whether it is limited in space were later extensively examined by the
philosopherImmanuelKantinhismonumental(andveryobscure)work
Critique of Pure Reason, published in 1781. He called these questions
antinomies(that is, contradictions)ofpure reason because hefelt that
there were equally compelling arguments for believing the thesis, that
the universe had a beginning, and the antithesis, that it had existed
forever.Hisargumentforthethesiswasthatiftheuniversedidnothave
abeginning,therewouldbeaninfiniteperiodoftimebeforeanyevent,
whichheconsideredabsurd.Theargumentfortheantithesiswasthatif
theuniversehadabeginning,therewouldbeaninfiniteperiodoftime
beforeit,sowhyshouldtheuniversebeginatanyoneparticulartime?
Infact,hiscasesforboththethesisandtheantithesisarereallythesame
argument. They are both based on his unspoken assumption that time
continuesbackforever,whetherornottheuniversehadexistedforever.
As we shall see, the concept of time has no meaning before the
beginningof the universe. This was firstpointed out by St. Augustine.
When asked: “What did God do before he created the universe?”
Augustine didn’t reply: “He was preparing Hell for people who asked
suchquestions.”Instead,hesaidthattimewasapropertyoftheuniverse
thatGodcreated,andthattimedidnotexistbeforethebeginningofthe
universe.
When most people believed in an essentially static and unchanging
universe,thequestionofwhether ornotithadabeginningwasreally
one of metaphysics or theology. One could account for what was
observedequallywellonthetheorythattheuniversehadexistedforever
oronthetheorythatitwassetinmotionatsomefinitetimeinsucha
mannerastolookasthoughithadexistedforever.Butin1929,Edwin
Hubblemadethelandmarkobservationthatwhereveryoulook,distant
galaxiesaremovingrapidlyawayfromus.Inotherwords,theuniverse
isexpanding.Thismeansthatatearliertimesobjectswouldhavebeen
closer together. In fact, it seemed that there was a time, about ten or
twenty thousand million years ago, when they were all at exactly the
sameplaceandwhen,therefore,thedensityoftheuniversewasinfinite.
This discovery finally brought the question of the beginning of the
universeintotherealmofscience.
Hubble’sobservationssuggestedthattherewasatime,calledthebig
bang,whentheuniversewasinfinitesimallysmallandinfinitelydense.
Undersuchconditionsallthelawsofscience,andthereforeallabilityto
predictthefuture,wouldbreakdown.Iftherewereeventsearlierthan
thistime,thentheycouldnotaffectwhathappensatthepresenttime.
Theirexistencecanbeignoredbecauseitwouldhavenoobservational
consequences.Onemaysaythattimehadabeginningatthebigbang,in
the sense that earlier times simply would not be defined. It should be
emphasizedthatthisbeginningintimeisverydifferentfromthosethat
hadbeenconsideredpreviously.Inanunchanginguniverseabeginning
intimeissomethingthathastobeimposedbysomebeingoutsidethe
universe;thereisnophysicalnecessityforabeginning.Onecanimagine
thatGod created the universeat literally any timein the past. On the
otherhand,iftheuniverseisexpanding,theremaybephysicalreasons
why there had to be a beginning. One could still imagine that God
createdtheuniverseattheinstantofthebigbang,orevenafterwardsin
justsuchawayastomakeitlookasthoughtherehadbeenabigbang,
butitwouldbemeaninglesstosupposethatitwascreatedbeforethebig
bang. An expanding universe does not preclude a creator, but it does
placelimitsonwhenhemighthavecarriedouthisjob!
In order to talk about the nature of the universe and to discuss
questionssuchaswhetherithasabeginningoranend,youhavetobe
clear about what a scientific theory is. I shall take the simpleminded
viewthatatheoryisjustamodeloftheuniverse,orarestrictedpartof
it,andasetofrulesthatrelatequantitiesinthemodeltoobservations
thatwemake.Itexistsonlyinourmindsanddoesnothaveanyother
reality (whatever that might mean). A theory is a good theory if it
satisfies two requirements. It must accurately describe a large class of
observationsonthebasisofamodelthatcontainsonlyafewarbitrary
elements, and it must make definite predictions about the results of
futureobservations.Forexample,AristotlebelievedEmpedocles’stheory
that everything was made out of four elements, earth, air, fire, and
water. This was simple enough, but did not make any definite
predictions.Ontheotherhand,Newton’stheoryofgravitywasbasedon
anevensimplermodel,inwhichbodiesattractedeachotherwithaforce
that was proportional to a quantity called their mass and inversely
proportionaltothesquareofthedistancebetweenthem.Yetitpredicts
themotionsofthe sun,themoon,and theplanetstoa highdegreeof
accuracy.
Anyphysicaltheoryisalwaysprovisional,inthesensethatitisonlya
hypothesis: you can never prove it. No matter how many times the
results of experiments agree with some theory, you can never be sure
thatthenexttimetheresultwillnotcontradictthetheory.Ontheother
hand, you can disprove a theory by finding even a single observation
that disagrees with the predictions of the theory. As philosopher of
scienceKarlPopperhasemphasized,agoodtheoryischaracterizedby
thefactthatitmakesanumberofpredictionsthatcouldinprinciplebe
disproved or falsified by observation. Each time new experiments are
observed to agree with the predictions the theory survives, and our
confidenceinitisincreased;butifeveranewobservationisfoundto
disagree,wehavetoabandonormodifythetheory.
At least that is what is supposed to happen, but you can always
questionthecompetenceofthepersonwhocarriedouttheobservation.
Inpractice,whatoftenhappensisthatanewtheoryisdevisedthatis
really an extension of the previous theory. For example, very accurate
observationsoftheplanetMercuryrevealedasmalldifferencebetween
itsmotionandthepredictionsofNewton’stheoryofgravity.Einstein’s
general theory of relativity predicted a slightly different motion from
Newton’stheory.ThefactthatEinstein’spredictionsmatchedwhatwas
seen,whileNewton’sdidnot,wasoneofthecrucialconfirmationsofthe
new theory. However, we still use Newton’s theory for all practical
purposes because the difference between its predictions and those of
general relativity is very small in the situations that we normally deal
with. (Newton’s theory also has the great advantage that it is much
simplertoworkwiththanEinstein’s!)
The eventual goal of science is to provide a single theory that
describes the whole universe. However, the approach most scientists
actuallyfollowistoseparatetheproblemintotwoparts.First,thereare
thelawsthattellushowtheuniversechangeswithtime.(Ifweknow
whattheuniverseislikeatanyonetime,thesephysicallawstellushow
itwilllookatanylatertime.)Second,thereisthequestionoftheinitial
stateoftheuniverse.Somepeoplefeelthatscienceshouldbeconcerned
withonlythefirstpart;theyregardthequestionoftheinitialsituation
asamatterformetaphysicsorreligion.TheywouldsaythatGod,being
omnipotent, could have started the universe off any way he wanted.
Thatmaybeso,butinthatcasehealsocouldhavemadeitdevelopina
completelyarbitraryway.Yetitappearsthathechosetomakeitevolve
in a very regular way according to certain laws. It therefore seems
equally reasonable to suppose that there are also laws governing the
initialstate.
It turns out to be very difficult to devise a theory to describe the
universeallinonego.Instead,webreaktheproblemupintobitsand
invent a number of partial theories. Each of these partial theories
describesandpredictsacertainlimitedclassofobservations,neglecting
the effects of other quantities, or representing them by simple sets of
numbers.Itmaybethatthisapproachiscompletelywrong.Ifeverything
in the universe depends on everything else in a fundamental way, it
mightbeimpossibletogetclosetoafullsolutionbyinvestigatingparts
oftheprobleminisolation.Nevertheless,itiscertainlythewaythatwe
have made progress in the past. The classic example again is the
Newtoniantheoryofgravity,whichtellsusthatthegravitationalforce
betweentwobodiesdependsonlyononenumberassociatedwitheach
body, its mass, but is otherwise independent of what the bodies are
madeof.Thusonedoesnotneedtohaveatheoryofthestructureand
constitutionofthesunandtheplanetsinordertocalculatetheirorbits.
Today scientists describe the universe in terms of two basic partial
theories—thegeneraltheoryofrelativityandquantummechanics.They
arethegreat intellectualachievementsof thefirsthalf ofthis century.
The general theory of relativity describes the force of gravity and the
large-scalestructureoftheuniverse,thatis,thestructureonscalesfrom
onlyafewmilestoaslargeasamillionmillionmillionmillion(1with
twenty-four zeros after it) miles, the size of the observable universe.
Quantum mechanics, on the other hand, deals with phenomena on
extremely small scales, such as a millionth of a millionth of an inch.
Unfortunately,however,thesetwotheoriesareknowntobeinconsistent
with each other—they cannot both be correct. One of the major
endeavors in physics today, and the major theme of this book, is the
search for a new theory that will incorporate them both—a quantum
theoryofgravity.Wedonotyethavesuchatheory,andwemaystillbe
a long way from having one, but we do already know many of the
propertiesthatitmusthave.Andweshallsee,inlaterchapters,thatwe
alreadyknowafairamountaboutthepredictionsaquantumtheoryof
gravitymustmake.
Now,ifyoubelievethattheuniverseisnotarbitrary,butisgoverned
bydefinitelaws,youultimatelyhavetocombinethepartialtheoriesinto
acompleteunifiedtheorythatwilldescribeeverythingintheuniverse.
But there is a fundamental paradox in the search for such a complete
unifiedtheory.Theideasaboutscientifictheoriesoutlinedaboveassume
wearerationalbeingswhoarefreetoobservetheuniverseaswewant
andtodrawlogicaldeductionsfromwhatwesee.Insuchaschemeitis
reasonable to suppose that we might progress ever closer toward the
lawsthatgovernouruniverse.Yetiftherereallyisacompleteunified
theory, it would also presumably determine our actions. And so the
theoryitselfwoulddeterminetheoutcomeofoursearchforit!Andwhy
should it determine that we come to the right conclusions from the
evidence?Mightitnotequallywelldeterminethatwedrawthewrong
conclusion?Ornoconclusionatall?
TheonlyanswerthatIcangivetothisproblemisbasedonDarwin’s
principleofnaturalselection.Theideaisthatinanypopulationofself-
reproducingorganisms,therewill bevariations inthegeneticmaterial
and upbringing that different individuals have. These differences will
meanthatsomeindividualsarebetterablethanotherstodrawtheright
conclusionsabouttheworldaroundthemandtoactaccordingly.These
individuals will be more likely to survive and reproduce and so their
patternofbehaviorandthoughtwillcometodominate.Ithascertainly
been true in the past that what we call intelligence and scientific
discoveryhaveconveyedasurvivaladvantage.Itisnotsoclearthatthis
is still the case: our scientific discoveries may well destroy us all, and
even if they don’t, a complete unified theory may not make much
difference to our chances of survival. However, provided the universe
has evolved in a regular way, we might expect that the reasoning
abilitiesthat natural selection has given us would be valid also in our
search for a complete unified theory, and so would not lead us to the
wrongconclusions.
Because the partial theories that we already have are sufficient to
make accurate predictions in all but the most extreme situations, the
searchfortheultimatetheoryoftheuniverseseemsdifficulttojustifyon
practical grounds. (It is worth noting, though, that similar arguments
could have been used against both relativity and quantum mechanics,
and these theories have given us both nuclear energy and the
microelectronicsrevolution!)Thediscoveryofacompleteunifiedtheory,
therefore,maynotaidthesurvivalofourspecies.Itmaynotevenaffect
our life-style. But ever since the dawn of civilization, people have not
beencontenttoseeeventsasunconnectedandinexplicable.Theyhave
cravedanunderstandingoftheunderlyingorderintheworld.Todaywe
still yearn to know why we are here and where we came from.
Humanity’sdeepestdesireforknowledgeisjustificationenoughforour
continuing quest. And our goal is nothing less than a complete
descriptionoftheuniversewelivein.
O
CHAPTER2
SPACEANDTIME
urpresentideasaboutthemotionofbodiesdatebacktoGalileoand
Newton. Before them people believed Aristotle, who said that the
naturalstateofabodywastobeatrestandthatitmovedonlyifdriven
bya force or impulse.It followed that aheavybody should fall faster
thanalightone,becauseitwouldhaveagreaterpulltowardtheearth.
The Aristotelian tradition also held that one could work out all the
lawsthatgoverntheuniversebypurethought:itwasnotnecessaryto
checkbyobservation.SonooneuntilGalileobotheredtoseewhether
bodies of different weight did in fact fall at different speeds. It is said
that Galileo demonstrated that Aristotle’s belief was false by dropping
weights from the leaning tower of Pisa. The story is almost certainly
untrue, but Galileo did do something equivalent: he rolled balls of
differentweightsdownasmoothslope.Thesituationissimilartothatof
heavy bodies falling vertically, but it is easier to observe because the
speeds are smaller. Galileo’s measurements indicated that each body
increased its speed at the same rate, no matter what its weight. For
example,ifyouletgoofaballonaslopethatdropsbyonemeterfor
everytenmetersyougoalong,theballwillbetravelingdowntheslope
ataspeedofaboutonemeterpersecondafteronesecond,twometers
per second after two seconds, and so on, however heavy the ball. Of
course a lead weight would fall faster than a feather, but that is only
because a feather is slowed down by air resistance. If one drops two
bodies that don’t have much air resistance, such as two different lead
weights,theyfallatthesamerate.Onthemoon,wherethereisnoairto
slow things down, the astronaut David R. Scott performed the feather
and lead weight experiment and found that indeed they did hit the
groundatthesametime.
Galileo’smeasurementswereusedbyNewtonasthebasisofhislaws
ofmotion.InGalileo’sexperiments,asabodyrolleddowntheslopeit
wasalwaysactedonbythesameforce(itsweight),andtheeffectwasto
makeitconstantlyspeedup.Thisshowedthattherealeffectofaforceis
alwaystochangethespeedofabody,ratherthanjusttosetitmoving,
as was previously thought. It also meant that whenever a body is not
actedonbyanyforce,it willkeep onmoving ina straightline atthe
same speed. This idea was first stated explicitly in Newton’s Principia
Mathematica, published in 1687, and is known as Newton’s first law.
WhathappenstoabodywhenaforcedoesactonitisgivenbyNewton’s
secondlaw.Thisstatesthatthebodywillaccelerate,orchangeitsspeed,
ataratethatisproportionaltotheforce.(Forexample,theacceleration
istwiceasgreatiftheforceistwiceasgreat.)Theaccelerationisalso
smallerthe greater the mass (or quantity of matter) of the body. (The
same force acting on a body of twice the mass will produce half the
acceleration.) A familiar example is provided by a car: the more
powerful the engine, the greater the acceleration, but the heavier the
car,thesmallertheaccelerationforthesameengine.Inadditiontohis
laws of motion, Newton discovered a law to describe the force of
gravity, which states that every body attracts every other body with a
force that is proportional to the mass of each body. Thus the force
betweentwobodieswouldbetwiceasstrongifoneofthebodies(say,
bodyA) had its mass doubled. This is what you might expect because
onecouldthinkofthenewbodyAasbeingmadeoftwobodieswiththe
originalmass.EachwouldattractbodyB withthe originalforce. Thus
thetotalforcebetweenAandBwouldbetwicetheoriginalforce.And
if,say, oneof the bodieshad twice themass, and theother had three
timesthemass,thentheforcewouldbesixtimesasstrong.Onecannow
seewhyallbodiesfallatthesamerate:abodyoftwicetheweightwill
have twice the force of gravity pulling it down, but it will also have
twicethemass.AccordingtoNewton’ssecondlaw,thesetwoeffectswill
exactly cancel each other, so the acceleration will be the same in all
cases.
Newton’slawofgravityalsotellsusthatthefartherapartthebodies,
thesmallertheforce.Newton’slawofgravitysaysthatthegravitational
attractionofastarisexactlyonequarterthatofasimilarstarathalfthe
distance. This law predicts the orbits of the earth, the moon, and the
planets with great accuracy. If the law were that the gravitational
attraction of a star went down faster or increased more rapidly with
distance, the orbits of the planets would not be elliptical, they would
eitherspiralintothesunorescapefromthesun.
ThebigdifferencebetweentheideasofAristotleandthoseofGalileo
andNewtonisthatAristotlebelievedinapreferredstateofrest,which
anybodywouldtakeupifitwerenotdrivenbysomeforceorimpulse.
Inparticular,hethoughtthattheearthwasatrest.Butitfollowsfrom
Newton’s laws that there is no unique standard of rest. One could
equally well say that body A was at rest and body B was moving at
constantspeedwithrespecttobodyA,orthatbodyB wasatrestand
body A was moving. For example, if one sets aside for a moment the
rotationoftheearthanditsorbitroundthesun,onecouldsaythatthe
earth was at rest and that a train on it was traveling north at ninety
milesperhour orthat thetrain wasat restandtheearthwasmoving
south at ninety miles per hour. If one carried out experiments with
moving bodies on the train, all Newton’s laws would still hold. For
instance, playing Ping-Pong on the train, one would find that the ball
obeyedNewton’slawsjustlikeaballonatablebythetrack.Sothereis
nowaytotellwhetheritisthetrainortheearththatismoving.
The lack of an absolute standard of rest meant that one could not
determinewhethertwoeventsthattookplaceatdifferenttimesoccurred
inthesamepositioninspace.Forexample,supposeourPing-Pongball
onthetrainbouncesstraightupanddown,hittingthetabletwiceonthe
samespotonesecondapart.Tosomeoneonthetrack,thetwobounces
would seem to take place about forty meters apart, because the train
wouldhavetraveledthatfardownthetrackbetweenthebounces.The
nonexistenceofabsoluterestthereforemeantthatonecouldnotgivean
event an absolute position in space, as Aristotle had believed. The
positionsofeventsandthe distancesbetweenthemwouldbedifferent
forapersononthetrainandoneonthetrack,andtherewouldbeno
reasontopreferoneperson’spositiontotheother’s.
Newtonwasveryworriedbythislackofabsoluteposition,orabsolute
space, as it was called, because it did not accord with his idea of an
absoluteGod.Infact,herefusedtoacceptlackofabsolutespace,even
though it was implied by his laws. He was severely criticized for this
irrational belief by many people, most notably by Bishop Berkeley, a
philosopher who believed that all material objects and space and time
are an illusion. When the famous Dr. Johnson was told of Berkeley’s
opinion,hecried,“Irefuteitthus!”andstubbedhistoeonalargestone.
Both Aristotle and Newton believed in absolute time. That is, they
believed that one could unambiguously measure the interval of time
between two events, and that this time would be the same whoever
measured it, provided they used a good clock. Time was completely
separatefromandindependentofspace.Thisiswhatmostpeoplewould
taketobethecommonsenseview.However,wehavehadtochangeour
ideas about space and time. Although our apparently commonsense
notionsworkwellwhendealingwiththingslikeapples,orplanetsthat
travelcomparativelyslowly,theydon’tworkatallforthingsmovingat
ornearthespeedoflight.
The fact that light travels at a finite, but very high, speed was first
discoveredin1676bytheDanishastronomerOleChristensenRoemer.
HeobservedthatthetimesatwhichthemoonsofJupiterappearedto
passbehindJupiterwerenotevenlyspaced,asonewouldexpectifthe
moonswentroundJupiterataconstantrate.AstheearthandJupiter
orbitaroundthesun,thedistancebetweenthemvaries.Roemernoticed
thateclipsesofJupiter’smoonsappearedlaterthefartherwewerefrom
Jupiter.Hearguedthatthiswasbecausethelightfromthemoonstook
longertoreachuswhenwewerefartheraway.Hismeasurementsofthe
variationsinthedistanceoftheearthfromJupiterwere,however,not
veryaccurate,andsohisvalueforthespeedoflightwas140,000miles
persecond,comparedtothemodernvalueof186,000milespersecond.
Nevertheless, Roemer’s achievement, in not only proving that light
travels at a finite speed, but also in measuring that speed, was
remarkable—comingasitdidelevenyearsbeforeNewton’spublication
ofPrincipiaMathematica.
A proper theory of the propagation of light didn’t come until 1865,
when the British physicist James Clerk Maxwell succeeded in unifying
thepartialtheoriesthatuptothenhadbeenusedtodescribetheforces
of electricity and magnetism. Maxwell’s equations predicted that there
could be wavelike disturbances in the combined electromagnetic field,
andthatthesewouldtravelatafixedspeed,likeripplesonapond.If
thewavelengthofthesewaves(thedistancebetweenonewavecrestand
thenext) isameter ormore, they arewhat we nowcall radio waves.
Shorter wavelengths are known as microwaves (a few centimeters) or
infrared(morethanaten-thousandthofacentimeter).Visiblelighthasa
wavelengthofbetweenonlyfortyandeightymillionthsofacentimeter.
Evenshorterwavelengthsareknownasultraviolet,Xrays,andgamma
rays.
Maxwell’stheorypredictedthatradioorlightwavesshouldtravelata
certain fixed speed. But Newton’s theory had got rid of the idea of
absolute rest, so if light was supposed to travel at a fixed speed, one
wouldhavetosaywhatthatfixedspeedwastobemeasuredrelativeto.
Itwasthereforesuggestedthattherewasasubstancecalledthe“ether”
thatwaspresenteverywhere,evenin“empty”space.Lightwavesshould
travel through the ether as sound waves travel through air, and their
speed should therefore be relative to the ether. Different observers,
moving relative to the ether, would see light coming toward them at
different speeds, but light’s speed relative to the ether would remain
fixed. In particular, as the earth was moving through the ether on its
orbitroundthesun,thespeedoflightmeasuredinthedirectionofthe
earth’s motion through the ether (when we were moving toward the
source of the light) should be higher than the speed of light at right
anglestothatmotion(whenwearenotmovingtowardthesource).In
1887AlbertMichelson(wholaterbecamethefirstAmericantoreceive
the Nobel Prize for physics) and Edward Morley carried out a very
carefulexperimentattheCaseSchoolofAppliedScienceinCleveland.
Theycomparedthespeedoflightinthedirectionoftheearth’smotion
withthatatright anglestotheearth’s motion.Totheirgreat surprise,
theyfoundtheywereexactlythesame!
Between1887and1905therewereseveralattempts,mostnotablyby
the Dutch physicist Hendrik Lorentz, to explain the result of the
Michelson-Morleyexperimentintermsofobjectscontractingandclocks
slowing down when they moved through the ether. However, in a
famous paper in 1905, a hitherto unknown clerk in the Swiss patent
office,AlbertEinstein,pointedoutthatthewholeideaofanetherwas
unnecessary,providingonewaswillingtoabandontheideaofabsolute
time.AsimilarpointwasmadeafewweekslaterbyaleadingFrench
mathematician, Henri Poincaré. Einstein’s arguments were closer to
physics than those of Poincaré, who regarded this problem as
mathematical.Einsteinisusuallygiventhecreditforthenewtheory,but
Poincaré is remembered by having his name attached to an important
partofit.
Thefundamentalpostulateofthetheoryofrelativity,asitwascalled,
was that the laws of science should be the same for all freely moving
observers,nomatterwhattheirspeed.ThiswastrueforNewton’slaws
ofmotion,butnowtheideawasextendedtoincludeMaxwell’stheory
andthespeedoflight:allobserversshouldmeasurethesamespeedof
light, no matter how fast they are moving. This simple idea has some
remarkableconsequences. Perhaps the best known are the equivalence
ofmassandenergy,summedupinEinstein’sfamousequationE=mc2
(whereEisenergy,mismass,andcisthespeedoflight),andthelaw
that nothing may travel faster than the speed of light. Because of the
equivalenceofenergyandmass,theenergywhichanobjecthasdueto
itsmotionwilladdtoitsmass.Inotherwords,itwillmakeitharderto
increaseitsspeed.Thiseffectisonlyreallysignificantforobjectsmoving
atspeedsclosetothespeedoflight.Forexample,at10percentofthe
speed of light an object’s mass is only 0.5 percent more than normal,
whileat90percentofthespeedoflightitwouldbemorethantwiceits
normalmass.Asanobjectapproachesthespeedoflight,itsmassrises
ever more quickly, so it takes more and more energy to speed it up
further.Itcaninfactneverreachthespeedoflight,becausebythenits
masswouldhavebecomeinfinite,andbytheequivalenceofmassand
energy,itwouldhavetakenaninfiniteamountofenergytogetitthere.
For this reason, any normal object is forever confined by relativity to
moveatspeedsslowerthanthespeedoflight.Onlylight,orotherwaves
thathavenointrinsicmass,canmoveatthespeedoflight.
An equally remarkable consequence of relativity is the way it has
revolutionizedourideasofspaceandtime.InNewton’stheory,ifapulse
of light is sent from one place to another, different observers would
agreeonthetimethatthejourneytook(sincetimeisabsolute),butwill
not always agree on how far the light traveled (since space is not
absolute).Sincethespeedofthelightisjustthedistanceithastraveled
divided by the time it has taken, different observers would measure
different speeds for the light. In relativity, on the other hand, all
observersmust agree on how fast light travels. They still, however, do
notagreeonthedistancethelighthastraveled,sotheymusttherefore
now also disagree over the time it has taken. (The time taken is the
distancethelighthastraveled—whichtheobserversdonotagreeon—
dividedbythelight’sspeed—whichtheydoagreeon.)Inotherwords,
the theory of relativity put an end to the idea of absolute time! It
appeared that each observer must have his own measure of time, as
recordedbyaclockcarriedwithhim,andthatidenticalclockscarried
bydifferentobserverswouldnotnecessarilyagree.
Eachobservercoulduseradartosaywhereandwhenaneventtook
placebysendingoutapulseoflightorradiowaves.Partofthepulseis
reflectedbackattheeventandtheobservermeasuresthetimeatwhich
hereceivestheecho.Thetimeoftheeventisthensaidtobethetime
halfway between when the pulse was sent and the time when the
reflectionwasreceivedback:the distanceoftheeventishalfthetime
takenforthisroundtrip,multipliedbythespeedoflight.(Anevent,in
thissense,issomethingthattakesplaceatasinglepointinspace,ata
specified point in time.) This idea is shown in Fig. 2.1, which is an
exampleofa space-timediagram.Using thisprocedure,observers who
are moving relative to each other will assign different times and
positionstothesameevent.Noparticularobserver’smeasurementsare
any more correct than any other observer’s, but all the measurements
arerelated.Anyobservercanworkoutpreciselywhattimeandposition
anyotherobserverwillassigntoanevent,providedheknowstheother
observer’srelativevelocity.
Nowadays we use just this method to measure distances precisely,
becausewecanmeasuretimemoreaccuratelythanlength.Ineffect,the
meter is defined to be the distance traveled by light in
0.000000003335640952 second, as measured by a cesium clock. (The
reasonforthatparticularnumberisthatitcorrespondstothehistorical
definitionofthemeter—intermsoftwomarksonaparticularplatinum
barkeptinParis.)Equally,wecanuseamoreconvenient,newunitof
lengthcalledalight-second.Thisissimplydefinedasthedistancethat
light travels in one second. In the theory of relativity, we now define
distance in terms of time and the speed of light, so it follows
automatically that every observer will measure light to have the same
speed (by definition, 1 meter per 0.000000003335640952 second).
There is no need to introduce the idea of an ether, whose presence
anyway cannot be detected, as the Michelson-Morley experiment
showed. The theory of relativity does, however, force us to change
fundamentallyourideasofspaceandtime.Wemustacceptthattimeis
notcompletelyseparatefromandindependentofspace,butiscombined
withittoformanobjectcalledspace-time.
FIGURE2.1Timeismeasuredvertically,andthedistancefromtheobserverismeasured
horizontally.Theobserver’spaththroughspaceandtimeisshownastheverticallineon
theleft.Thepathsoflightraystoandfromtheeventarethediagonallines.
Itisamatterofcommonexperiencethatonecandescribetheposition
ofapointinspacebythreenumbers,orcoordinates.Forinstance,one
can say that a point in a room is seven feet from one wall, three feet
fromanother,andfivefeetabovethefloor.Oronecouldspecifythata
pointwasatacertainlatitudeandlongitudeandacertainheightabove
sealevel.Oneisfreetouseanythreesuitablecoordinates,althoughthey
haveonlyalimitedrangeofvalidity.Onewouldnotspecifytheposition
ofthemoonintermsofmilesnorthandmileswestofPiccadillyCircus
and feet above sea level. Instead, one might describe it in terms of
distance from the sun, distance from the plane of the orbits of the
planets,andtheanglebetweenthelinejoiningthemoontothesunand
theline joining the sunto a nearby starsuchas Alpha Centauri.Even
thesecoordinateswouldnotbeofmuchuseindescribingthepositionof
thesuninourgalaxyorthepositionofourgalaxyinthelocalgroupof
galaxies. In fact, one may describe the whole universe in terms of a
collectionofoverlappingpatches.Ineachpatch,onecanuseadifferent
setofthreecoordinatestospecifythepositionofapoint.
Aneventissomethingthathappensataparticularpointinspaceand
at a particular time. So one can specify it by four numbers or
coordinates. Again, the choice of coordinates is arbitrary; one can use
anythreewell-definedspatialcoordinatesandanymeasureoftime.In
relativity, there is no real distinction between the space and time
coordinates, just as there is no real difference between any two space
coordinates.One could choose a new setof coordinates in which, say,
thefirstspacecoordinatewasacombinationoftheoldfirstandsecond
spacecoordinates. For instance,instead of measuring thepositionof a
point on the earth in miles north of Piccadilly and miles west of
Piccadilly, one could use miles northeast of Piccadilly, and miles
northwestofPiccadilly.Similarly,inrelativity,onecoulduseanewtime
coordinatethatwastheoldtime(inseconds)plusthedistance(inlight-
seconds)northofPiccadilly.
It is often helpful to think of the four coordinates of an event as
specifyingitspositioninafour-dimensionalspacecalledspace-time.Itis
impossibletoimagineafour-dimensionalspace.Ipersonallyfindithard
enoughtovisualizethree-dimensionalspace!However,itiseasytodraw
diagrams of two-dimensional spaces, such as the surface of the earth.
(The surface of the earth is two-dimensional because the position of a
pointcanbespecifiedbytwocoordinates,latitudeandlongitude.)Ishall
generallyusediagramsinwhichtimeincreasesupwardandoneofthe
spatial dimensions is shown horizontally. The other two spatial
dimensions are ignored or, sometimes, one of them is indicated by
perspective. (These are called space-time diagrams, like Fig. 2.1.) For
example,inFig.2.2timeismeasuredupwardinyearsandthedistance
alongthelinefromthesuntoAlphaCentauriismeasuredhorizontally
inmiles.ThepathsofthesunandofAlphaCentaurithroughspace-time
areshownastheverticallinesontheleftandrightofthediagram.Aray
oflightfromthesunfollowsthediagonalline,andtakesfouryearsto
getfromthesuntoAlphaCentauri.
Aswehaveseen,Maxwell’sequationspredictedthatthespeedoflight
shouldbethesamewhateverthespeedofthesource,andthishasbeen
confirmedbyaccuratemeasurements.Itfollowsfromthisthatifapulse
oflightisemittedataparticulartimeataparticularpointinspace,then
as time goes on it will spread out as a sphere of light whose size and
positionareindependentofthespeedofthesource.Afteronemillionth
ofasecondthelightwillhavespreadouttoformaspherewitharadius
of300meters;aftertwomillionthsofasecond,theradiuswillbe600
meters; and so on. It will be like the ripples that spread out on the
surfaceofapondwhenastoneisthrownin.Theripplesspreadoutasa
circle that gets bigger as time goes on. If one stacks snapshots of the
ripples at different times one above the other, the expanding circle of
rippleswillmarkoutaconewhosetipisattheplaceandtimeatwhich
thestonehitthewater(Fig.2.3).Similarly,thelightspreadingoutfrom
an event forms a (three-dimensional) cone in (the four-dimensional)
space-time.Thisconeiscalledthefuturelightconeoftheevent.Inthe
samewaywecandrawanothercone,calledthepastlightcone,whichis
thesetofeventsfromwhichapulseoflightisabletoreachthegiven
event(Fig.2.4).
FIGURE2.2
FIGURE2.3
FIGURE2.4
GivenaneventP,onecandividetheothereventsintheuniverseinto
three classes. Those events that can be reached from theevent P by a
particleorwavetravelingatorbelowthespeedoflightaresaidtobein
thefutureofP.Theywillliewithinorontheexpandingsphereoflight
emittedfromtheeventP.Thustheywillliewithinoronthefuturelight
coneofPinthespace-timediagram.OnlyeventsinthefutureofPcan
beaffectedbywhathappensatPbecausenothingcantravelfasterthan
light.
Similarly, the past of P can be defined as the set of all events from
whichitispossibletoreachtheeventPtravelingatorbelowthespeed
oflight.ItisthusthesetofeventsthatcanaffectwhathappensatP.The
events that do not lie in the future or past of P are said to lie in the
elsewhereofP(Fig.2.5).Whathappensatsucheventscanneitheraffect
norbeaffectedbywhathappensatP.Forexample,ifthesunwereto
ceasetoshineatthisverymoment,itwouldnotaffectthingsonearthat
the present time because they would be in the elsewhere of the event
when the sun went out (Fig.2.6). We would know about it only after
eightminutes,thetimeittakeslighttoreachusfromthesun.Onlythen
wouldeventsonearthlieinthefuturelightconeoftheeventatwhich
thesunwentout.Similarly,wedonotknowwhatishappeningatthe
momentfartherawayintheuniverse:thelightthatweseefromdistant
galaxies left them millions of years ago, and in the case of the most
distant object that we have seen, the light left some eight thousand
millionyearsago.Thus,whenwelookattheuniverse,weareseeingit
asitwasinthepast.
FIGURE2.5
If one neglects gravitational effects, as Einstein and Poincaré did in
1905,onehaswhatiscalledthespecialtheoryofrelativity.Forevery
eventinspace-timewemayconstructalightcone(thesetofallpossible
pathsoflightinspace-timeemittedatthatevent),andsincethespeedof
lightisthesameateveryeventandineverydirection,allthelightcones
willbeidenticalandwillallpointinthesamedirection.Thetheoryalso
tellsusthatnothingcantravelfasterthanlight.Thismeansthatthepath
ofanyobjectthroughspaceandtimemustberepresentedbyalinethat
lies within the light cone at each event on it (Fig. 2.7). The special
theoryof relativitywas verysuccessfulin explainingthat thespeedof
light appears the same to all observers (as shown by the Michelson-
Morleyexperiment)andindescribingwhathappenswhenthingsmove
atspeedsclosetothespeedoflight.However,itwasinconsistentwith
theNewtoniantheoryofgravity,whichsaidthatobjectsattractedeach
other with a force that depended on the distance between them. This
meantthatifonemovedoneoftheobjects,theforceontheotherone
would change instantaneously. Or in other words, gravitational effects
shouldtravelwithinfinitevelocity,insteadofatorbelowthespeedof
light, as the special theory of relativity required. Einstein made a
number of unsuccessful attempts between 1908 and 1914 to find a
theoryofgravitythatwasconsistentwithspecial relativity.Finally,in
1915,heproposedwhatwenowcallthegeneraltheoryofrelativity.
FIGURE2.6
FIGURE2.7
Einsteinmadetherevolutionarysuggestionthatgravityisnotaforce
likeotherforces,butisaconsequenceofthefactthatspace-timeisnot
flat,ashadbeenpreviouslyassumed:itiscurved,or“warped,”bythe
distributionofmassandenergyinit.Bodiesliketheeartharenotmade
tomoveoncurvedorbitsbyaforcecalledgravity;instead,theyfollow
thenearestthingtoastraightpathinacurvedspace,whichiscalleda
geodesic. A geodesic is the shortest (or longest) path between two
nearbypoints.Forexample,thesurfaceoftheearthisatwo-dimensional
curvedspace.Ageodesicontheearthiscalledagreatcircle,andisthe
shortest route between two points (Fig. 2.8). As the geodesic is the
shortest path between any two airports, this is the route an airline
navigator will tell the pilot to fly along. In general relativity, bodies
always follow straight lines in four-dimensional space-time, but they
nevertheless appear to us to move along curved paths in our three-
dimensionalspace.(Thisisratherlikewatchinganairplaneflyingover
hilly ground. Although it follows a straight line in three-dimensional
space,itsshadowfollowsacurvedpathonthetwo-dimensionalground.)
FIGURE2.8
Themassofthesuncurvesspace-timeinsuchawaythatalthoughthe
earthfollowsastraightpathinfour-dimensionalspace-time,itappears
toustomovealongacircularorbitinthree-dimensionalspace.Infact,
the orbits of the planets predicted by general relativity are almost
exactlythesameasthosepredictedbytheNewtoniantheoryofgravity.
However,inthecaseofMercury,which,beingthenearestplanettothe
sun,feelsthestrongestgravitationaleffects,andhasaratherelongated
orbit,generalrelativitypredictsthatthelongaxisoftheellipseshould
rotateaboutthesunatarateofaboutonedegreeintenthousandyears.
Smallthoughthiseffectis,ithadbeennoticedbefore1915andserved
asoneofthefirstconfirmationsofEinstein’stheory.Inrecentyearsthe
even smaller deviations of the orbits of the other planets from the
Newtonianpredictionshavebeenmeasuredbyradarandfoundtoagree
withthepredictionsofgeneralrelativity.
Lightraystoomustfollowgeodesicsinspace-time.Again,thefactthat
spaceiscurvedmeansthatlightnolongerappearstotravelinstraight
linesinspace.Sogeneralrelativitypredictsthatlightshouldbebentby
gravitationalfields.Forexample,thetheorypredictsthatthelightcones
ofpointsnearthesunwouldbeslightlybentinward,onaccountofthe
massofthesun.Thismeansthatlightfromadistantstarthathappened
topassnearthesunwouldbedeflectedthroughasmallangle,causing
thestartoappearinadifferentpositiontoanobserverontheearth(Fig.
2.9).Ofcourse,ifthelightfromthestaralwayspassedclosetothesun,
wewouldnotbeabletotellwhetherthelightwasbeingdeflectedorif
insteadthestarwasreallywhereweseeit.However,astheearthorbits
aroundthesun,differentstarsappeartopassbehindthesunandhave
their light deflected. They therefore change their apparent position
relativetootherstars.
FIGURE2.9
Itisnormallyverydifficult tosee thiseffect, becausethe lightfrom
thesunmakesitimpossibletoobservestarsthatappearneartothesun
inthesky.However,itispossibletodosoduringaneclipseofthesun,
whenthesun’slightisblockedoutbythemoon.Einstein’spredictionof
light deflection could not be tested immediately in 1915, because the
FirstWorldWarwasinprogress,anditwasnotuntil1919thataBritish
expedition,observinganeclipsefromWestAfrica,showedthatlightwas
indeeddeflectedbythesun,justaspredictedbythetheory.Thisproof
of a German theory by British scientists was hailed as a great act of
reconciliation between the two countries after the war. It is ironic,
therefore, that later examination of the photographs taken on that
expeditionshowedtheerrorswereasgreatastheeffecttheyweretrying
to measure. Their measurement had been sheer luck, or a case of
knowingtheresulttheywantedtoget,notanuncommonoccurrencein
science.Thelightdeflectionhas,however,beenaccuratelyconfirmedby
anumberoflaterobservations.
Anotherpredictionofgeneralrelativityisthattimeshouldappearto
runslowernearamassivebodyliketheearth.Thisisbecausethereisa
relation between the energy of light and its frequency (that is, the
numberofwavesoflightpersecond):thegreatertheenergy,thehigher
thefrequency.Aslighttravelsupwardintheearth’sgravitationalfield,it
loses energy, and so its frequency goes down. (This means that the
length of time between one wave crest and the next goes up.) To
someone high up, it would appear that everything down below was
takinglongertohappen.Thispredictionwastestedin1962,usingapair
ofveryaccurateclocksmountedatthetopandbottomofawatertower.
Theclockatthebottom,whichwasnearertheearth,wasfoundtorun
slower,inexactagreementwithgeneralrelativity.Thedifferenceinthe
speed of clocks at different heights above the earth is now of
considerable practical importance, with the advent of very accurate
navigation systems based on signals from satellites. If one ignored the
predictionsofgeneralrelativity,thepositionthatonecalculatedwould
bewrongbyseveralmiles!
Newton’slawsofmotionputanendtotheideaofabsolutepositionin
space.Thetheoryofrelativitygetsridofabsolutetime.Considerapair
oftwins.Suppose thatone twingoes toliveonthetop ofa mountain
whiletheotherstaysatsealevel.Thefirsttwinwouldagefasterthan
thesecond.Thus,iftheymetagain,onewouldbeolderthantheother.
Inthiscase,thedifferenceinageswouldbeverysmall,butitwouldbe
much larger if one of the twins went for a long trip in a spaceship at
nearlythespeedoflight.Whenhereturned,hewouldbemuchyounger
thantheonewhostayedonearth.Thisisknownasthetwinsparadox,
butitisaparadoxonlyifonehastheideaofabsolutetimeattheback
of one’s mind. In the theory of relativity there is no unique absolute
time,butinsteadeachindividualhashisownpersonalmeasureoftime
thatdependsonwhereheisandhowheismoving.
Before1915,spaceandtimewerethoughtofasafixedarenainwhich
eventstookplace,butwhichwasnotaffectedbywhathappenedinit.
This was true even of the special theory of relativity. Bodies moved,
forces attracted and repelled, but time and space simply continued,
unaffected.Itwasnaturaltothinkthatspaceandtimewentonforever.
The situation, however, is quite different in the general theory of
relativity. Space and time are now dynamic quantities: when a body
moves,oraforceacts,itaffectsthecurvatureofspaceandtime—andin
turn the structure of space-time affects the way in which bodies move
andforcesact.Spaceandtimenotonlyaffectbutalsoareaffectedby
everythingthathappensintheuniverse.Justasonecannottalkabout
events in the universe without the notions of space and time, so in
general relativity it became meaningless to talk about space and time
outsidethelimitsoftheuniverse.
In the following decades this new understanding of space and time
was to revolutionize our view of the universe. The old idea of an
essentially unchanging universe that could have existed, and could
continue to exist, forever was replaced by the notion of a dynamic,
expanding universe that seemed to have begun a finite time ago, and
thatmightendatafinitetimeinthefuture.Thatrevolutionformsthe
subjectofthenextchapter.Andyearslater,itwasalsotobethestarting
point for my work in theoretical physics. Roger Penrose and I showed
thatEinstein’sgeneraltheoryofrelativityimpliedthattheuniversemust
haveabeginningand,possibly,anend.
I
CHAPTER3
THEEXPANDING
UNIVERSE
fonelooksattheskyonaclear,moonlessnight,thebrightestobjects
oneseesarelikelytobetheplanetsVenus,Mars,Jupiter,andSaturn.
Therewillalsobeaverylargenumberofstars,whicharejustlikeour
ownsunbutmuchfartherfromus.Someofthesefixedstarsdo,infact,
appearto changevery slightlytheir positions relativeto eachotheras
theearthorbitsaroundthesun:theyarenotreallyfixedatall!Thisis
becausetheyarecomparativelyneartous.Astheearthgoesroundthe
sun, we see them from different positions against the background of
more distant stars. This is fortunate, because it enables us to measure
directlythedistanceofthesestarsfromus:thenearertheyare,themore
theyappeartomove.Theneareststar,calledProximaCentauri,isfound
to be about four light-years away (the light from it takes about four
yearstoreachearth),orabouttwenty-threemillionmillionmiles.Most
of the other stars that are visible to the naked eye lie within a few
hundredlight-yearsofus.Oursun,forcomparison,isamereeightlight-
minutesaway!Thevisiblestarsappearspreadalloverthenightsky,but
areparticularlyconcentratedinoneband,whichwecalltheMilkyWay.
As long ago as 1750, some astronomers were suggesting that the
appearanceoftheMilkyWaycouldbeexplainedifmostofthevisible
starslieinasingledisklikeconfiguration,oneexampleofwhatwenow
callaspiralgalaxy.Onlyafewdecadeslater,theastronomerSirWilliam
Herschel confirmed this idea by painstakingly cataloging the positions
anddistancesofvastnumbersofstars.Evenso,theideagainedcomplete
acceptanceonlyearlythiscentury.
Ourmodernpictureoftheuniversedatesbacktoonly1924,whenthe
AmericanastronomerEdwinHubbledemonstratedthatourswasnotthe
onlygalaxy.Therewereinfactmanyothers,withvasttractsofempty
spacebetweenthem.Inordertoprovethis,heneededtodeterminethe
distances to these other galaxies, which are so far away that, unlike
nearbystars,theyreallydoappearfixed.Hubblewasforced,therefore,
to use indirect methods to measure the distances. Now, the apparent
brightnessofastardependsontwofactors:howmuchlightitradiates
(its luminosity), and how far it is from us. For nearby stars, we can
measure their apparent brightness and their distance, and so we can
workouttheirluminosity.Conversely,ifweknewtheluminosityofstars
inothergalaxies,wecouldworkouttheirdistancebymeasuringtheir
apparent brightness. Hubble noted that certain types of stars always
havethesameluminositywhentheyarenearenoughforustomeasure;
therefore,heargued,ifwefoundsuchstarsinanothergalaxy,wecould
assumethattheyhadthesameluminosity—andsocalculatethedistance
to that galaxy. If we could do this for a number of stars in the same
galaxy,andourcalculationsalwaysgavethesamedistance,wecouldbe
fairlyconfidentofourestimate.
Inthisway,EdwinHubbleworkedoutthedistancestoninedifferent
galaxies. We now know that our galaxy is only one of some hundred
thousandmillionthatcanbeseenusingmoderntelescopes,eachgalaxy
itselfcontainingsomehundredthousandmillionstars.Fig.3.1showsa
pictureofonespiralgalaxythatissimilartowhatwethinkoursmust
lookliketosomeonelivinginanothergalaxy.Weliveinagalaxythatis
about one hundred thousand light-years across and is slowly rotating;
the stars in its spiral arms orbit around its center about once every
severalhundredmillionyears.Oursunisjustanordinary,average-sized,
yellow star, near the inner edge of one of the spiral arms. We have
certainlycomealongwaysinceAristotleandPtolemy,whenwethought
thattheearthwasthecenteroftheuniverse!
FIGURE3.1
Stars are so far away that they appear to us to be just pinpoints of
light. We cannot see their size or shape. So how can we tell different
types of stars apart? For the vast majority of stars, there is only one
characteristic feature that we can observe—the color of their light.
Newtondiscoveredthatiflightfromthesunpassesthroughatriangular-
shaped piece of glass, called a prism, it breaks up into its component
colors (its spectrum) as in a rainbow. By focusing a telescope on an
individualstarorgalaxy,onecansimilarlyobservethespectrumofthe
lightfromthatstarorgalaxy.Differentstarshavedifferentspectra,but
therelativebrightnessofthedifferentcolorsisalwaysexactlywhatone
wouldexpecttofindinthelightemittedbyanobjectthatisglowingred
hot.(Infact,thelightemittedbyanyopaqueobjectthatisglowingred
hothasacharacteristicspectrumthatdependsonlyonitstemperature—
a thermal spectrum. This means that we can tell a star’s temperature
from the spectrum of its light.) Moreover, we find that certain very
specificcolorsaremissingfromstars’spectra,andthesemissingcolors
mayvaryfromstartostar.Sinceweknowthateachchemicalelement
absorbsacharacteristicsetofveryspecificcolors,bymatchingtheseto
thosethataremissingfromastar’sspectrum,wecandetermineexactly
whichelementsarepresentinthestar’satmosphere.
Inthe1920s,whenastronomersbegantolookatthespectraofstarsin
othergalaxies,theyfoundsomethingmostpeculiar:therewerethesame
characteristicsetsofmissingcolorsasforstarsinourowngalaxy,but
theywereallshiftedbythesamerelativeamounttowardtheredendof
the spectrum. To understand the implications of this, we must first
understandtheDopplereffect.Aswehaveseen,visiblelightconsistsof
fluctuations,orwaves,intheelectromagneticfield.Thewavelength(or
distance from one wave crest to the next) of light is extremely small,
ranging from four to seven ten-millionths of a meter. The different
wavelengths of light are what the human eye sees as different colors,
withthelongestwavelengthsappearingattheredendofthespectrum
andtheshortestwavelengthsattheblueend.Nowimagineasourceof
light at a constant distance from us, such as a star, emitting waves of
lightataconstantwavelength.Obviouslythewavelengthofthewaves
wereceivewillbethesameasthewavelengthatwhichtheyareemitted
(thegravitationalfieldofthegalaxywillnotbelargeenoughtohavea
significanteffect).Supposenowthatthesourcestartsmovingtowardus.
Whenthesourceemitsthenextwavecrestitwillbenearertous,sothe
distance between wave crests will be smaller than when the star was
stationary.This means that the wavelengthof the waves we receive is
shorterthanwhenthestarwasstationary.Correspondingly,ifthesource
ismovingawayfromus,thewavelengthofthewaveswereceivewillbe
longer.Inthecaseoflight,therefore,thismeansthatstarsmovingaway
from us will have their spectra shifted toward the red end of the
spectrum (red-shifted) and those moving toward us will have their
spectra blue-shifted. This relationship between wavelength and speed,
whichiscalledtheDopplereffect,isaneverydayexperience.Listentoa
carpassingontheroad:asthecarisapproaching,itsenginesoundsata
higher pitch (corresponding to a shorter wavelength and higher
frequencyofsoundwaves),andwhenitpassesandgoesaway,itsounds
atalowerpitch.Thebehavioroflightorradiowavesissimilar.Indeed,
thepolicemakeuseoftheDopplereffecttomeasurethespeedofcarsby
measuringthewavelengthofpulsesofradiowavesreflectedoffthem.
In the years following his proof of the existence of other galaxies,
Hubble spent his time cataloging their distances and observing their
spectra. At that time most people expected the galaxies to be moving
around quite randomly, and so expected to find as many blue-shifted
spectraasred-shiftedones.Itwasquiteasurprise,therefore,tofindthat
most galaxies appeared red-shifted: nearly all were moving away from
us!MoresurprisingstillwasthefindingthatHubblepublishedin1929:
even the size of a galaxy’s red shift is not random, but is directly
proportional to the galaxy’s distance from us. Or, in other words, the
fartheragalaxyis,thefasteritismovingaway!Andthatmeantthatthe
universecouldnotbestatic,aseveryonepreviouslyhadthought,butis
infactexpanding;thedistancebetweenthedifferentgalaxiesisgrowing
allthetime.
The discovery that the universe is expanding was one of the great
intellectual revolutions of the twentieth century. With hindsight, it is
easy to wonder why no one had thought of it before. Newton, and
others, should have realized that a static universe would soon start to
contract under the influence of gravity. But suppose instead that the
universe is expanding. If it was expanding fairly slowly, the force of
gravity would cause it eventually to stop expanding and then to start
contracting.However,ifitwasexpandingatmorethanacertaincritical
rate,gravitywouldneverbestrongenoughtostopit,andtheuniverse
wouldcontinuetoexpandforever.Thisisabitlikewhathappenswhen
onefiresarocketupwardfromthesurfaceoftheearth.Ifithasafairly
lowspeed,gravitywilleventuallystoptherocketanditwillstartfalling
back.Ontheotherhand,ifthe rockethasmorethanacertaincritical
speed(aboutsevenmilespersecond),gravitywillnotbestrongenough
topullitback,soitwillkeepgoingawayfromtheearthforever.This
behavior of the universe could have been predicted from Newton’s
theoryofgravityatanytimeinthenineteenth,theeighteenth,oreven
the late seventeenth century. Yet so strong was the belief in a static
universethatitpersistedintotheearlytwentiethcentury.EvenEinstein,
whenheformulatedthegeneraltheoryofrelativityin1915,wassosure
thattheuniversehadtobe staticthathe modifiedhistheorytomake
this possible, introducing a so-called cosmological constant into his
equations.Einsteinintroducedanew“antigravity”force,which,unlike
otherforces,didnotcomefromanyparticularsourcebutwasbuiltinto
theveryfabricofspace-time.Heclaimedthatspace-timehadaninbuilt
tendency to expand, and this could be made to balance exactly the
attraction of all the matter in the universe, so that a static universe
would result. Only one man, it seems, was willing to take general
relativity at face value, and while Einstein and other physicists were
lookingforwaysofavoidinggeneralrelativity’spredictionofanonstatic
universe,theRussianphysicistandmathematicianAlexanderFriedmann
insteadsetaboutexplainingit.
Friedmannmadetwoverysimpleassumptionsabouttheuniverse:that
theuniverselooksidenticalinwhicheverdirectionwelook,andthatthis
would also be true if we were observing the universe from anywhere
else.Fromthesetwoideasalone,Friedmannshowedthatweshouldnot
expect the universe to be static. In fact, in 1922, several years before
Edwin Hubble’s discovery, Friedmann predicted exactly what Hubble
found!
Theassumptionthattheuniverselooksthesameineverydirectionis
clearlynottrueinreality.Forexample,aswehaveseen,theotherstars
inourgalaxyformadistinctbandoflight acrossthenight sky,called
theMilkyWay.Butifwelookatdistantgalaxies,thereseemstobemore
or less the same number of them. So the universe does seem to be
roughly the same in every direction, provided one views it on a large
scale compared to the distance between galaxies, and ignores the
differences on small scales. For a long time, this was sufficient
justification for Friedmann’s assumption—as a rough approximation to
therealuniverse.Butmorerecentlyaluckyaccidentuncoveredthefact
thatFriedmann’sassumptionisinfactaremarkablyaccuratedescription
ofouruniverse.
In1965twoAmericanphysicistsattheBellTelephoneLaboratoriesin
New Jersey, Arno Penzias and Robert Wilson, were testing a very
sensitivemicrowavedetector.(Microwavesarejustlikelightwaves,but
with a wavelength of around a centimeter.) Penzias and Wilson were
worriedwhentheyfoundthattheirdetectorwaspickingupmorenoise
than it ought to. The noise did not appear to be coming from any
particular direction. First they discovered bird droppings in their
detector and checked for other possible malfunctions, but soon ruled
theseout.Theyknewthatanynoisefromwithintheatmospherewould
bestrongerwhenthedetectorwasnotpointingstraightupthanwhenit
was, because light rays travel through much more atmosphere when
received from near the horizon than when received from directly
overhead.Theextranoisewasthesamewhicheverdirectionthedetector
waspointed,so itmust comefrom outsidethe atmosphere.It wasalso
thesamedayandnightandthroughouttheyear,eventhoughtheearth
wasrotatingonitsaxisandorbitingaroundthesun.Thisshowedthat
theradiationmustcomefrombeyondtheSolarSystem,andevenfrom
beyondthegalaxy,asotherwiseitwouldvaryasthemovementofearth
pointedthedetectorindifferentdirections.
In fact, we know that the radiation must have traveled to us across
mostoftheobservableuniverse,andsinceitappearstobethesamein
different directions, the universe must also be the same in every
direction, if only on a large scale. We now know that whichever
directionwelook,thisnoisenevervariesbymorethanatinyfraction:
so Penzias and Wilson had unwittingly stumbled across a remarkably
accurate confirmation of Friedmann’s first assumption. However,
becausetheuniverseisnotexactlythesameineverydirection,butonly
onaverageonalargescale,themicrowavescannotbeexactlythesame
in every direction either. There have to be slight variations between
different directions. These were first detected in 1992 by the Cosmic
BackgroundExplorersatellite,orCOBE,atalevelofaboutonepartina
hundred thousand. Small though these variations are, they are very
important,aswillbeexplainedinChapter8.
At roughly the same time as Penzias and Wilson were investigating
noise in their detector, two American physicists at nearby Princeton
University,Bob Dickeand JimPeebles, were alsotaking aninterestin
microwaves. They were working on a suggestion, made by George
Gamow(onceastudentofAlexanderFriedmann),thattheearlyuniverse
should have been very hot and dense, glowing white hot. Dicke and
Peeblesarguedthatweshouldstillbeabletoseetheglowoftheearly
universe,becauselightfromverydistantpartsofitwouldonlyjustbe
reachingusnow.However,theexpansionoftheuniversemeantthatthis
lightshouldbesogreatlyred-shiftedthatitwouldappeartousnowas
microwaveradiation.DickeandPeebleswerepreparingtolookforthis
radiationwhenPenziasandWilsonheardabouttheirworkandrealized
that they had already found it. For this, Penzias and Wilson were
awardedtheNobelPrizein1978(whichseemsabithardonDickeand
Peebles,nottomentionGamow!).
Nowatfirstsight,allthisevidencethattheuniverselooksthesame
whicheverdirectionwelookinmightseemtosuggestthereissomething
specialaboutourplaceintheuniverse.Inparticular,itmightseemthat
if we observe all other galaxies to be moving away from us, then we
must be at the center of the universe. There is, however, an alternate
explanation:theuniversemightlookthesameineverydirectionasseen
from any other galaxy too. This, as we have seen, was Friedmann’s
secondassumption.Wehavenoscientificevidencefor,oragainst,this
assumption.Webelieveitonlyongroundsofmodesty:itwouldbemost
remarkableiftheuniverselookedthesameineverydirectionaroundus,
butnotaroundotherpointsintheuniverse!InFriedmann’smodel,all
thegalaxiesaremovingdirectlyawayfromeachother.Thesituationis
ratherlikeaballoonwithanumberofspotspaintedonitbeingsteadily
blownup.Astheballoonexpands,thedistancebetweenanytwospots
increases,butthere isnospot thatcan besaid tobe thecenter ofthe
expansion.Moreover,thefartherapartthespotsare,thefastertheywill
be moving apart. Similarly, in Friedmann’s model the speed at which
any two galaxies are moving apart is proportional to the distance
betweenthem. So it predictedthat the red shiftofa galaxy should be
directly proportional to its distance from us, exactly as Hubble found.
Despite the success of his model and his prediction of Hubble’s
observations,Friedmann’sworkremainedlargelyunknownintheWest
untilsimilarmodelswerediscoveredin1935bytheAmericanphysicist
Howard Robertson and the British mathematician Arthur Walker, in
responsetoHubble’sdiscoveryoftheuniformexpansionoftheuniverse.
AlthoughFriedmannfoundonlyone,thereareinfactthreedifferent
kindsofmodelsthatobeyFriedmann’stwofundamentalassumptions.In
the first kind (which Friedmann found) the universe is expanding
sufficientlyslowlythatthegravitationalattractionbetweenthedifferent
galaxiescausestheexpansiontoslowdownandeventuallytostop.The
galaxies then start to move toward each other and the universe
contracts. Fig. 3.2 shows how the distance between two neighboring
galaxies changes as time increases. It starts at zero, increases to a
maximum, and then decreases to zero again. In the second kind of
solution, the universe is expanding so rapidly that the gravitational
attractioncanneverstopit,thoughitdoesslowitdownabit.Fig.3.3
shows the separation between neighboring galaxies in this model. It
startsatzeroandeventuallythegalaxiesaremovingapartatasteady
speed.Finally,thereisathirdkindofsolution,inwhichtheuniverseis
expanding only just fast enough to avoid recollapse. In this case the
separation,showninFig.3.4,also startsatzeroandincreases forever.
However,thespeedatwhichthegalaxiesaremovingapartgetssmaller
andsmaller,althoughitneverquitereacheszero.
FIGURE3.2
FIGURE3.3
FIGURE3.4
AremarkablefeatureofthefirstkindofFriedmannmodelisthatinit
the universe is not infinite in space, but neither does space have any
boundary. Gravity is so strong that space is bent round onto itself,
makingitratherlikethesurfaceoftheearth.Ifonekeepstravelingina
certaindirectiononthesurfaceoftheearth,onenevercomesupagainst
animpassablebarrierorfallsovertheedge,buteventuallycomesback
towhereonestarted.InthefirstFriedmannmodel,spaceisjustlikethis,
but with three dimensions instead of two for the earth’s surface. The
fourthdimension,time,isalsofiniteinextent,butitislikealinewith
twoendsorboundaries,abeginningandanend.Weshallseelaterthat
whenonecombinesgeneralrelativitywiththeuncertaintyprincipleof
quantummechanics, it is possible for both space and time to be finite
withoutanyedgesorboundaries.
Theideathatonecouldgorightroundtheuniverseandendupwhere
one started makes good science fiction, but it doesn’t have much
practicalsignificance,becauseitcanbeshownthattheuniversewould
recollapsetozerosizebeforeonecouldgetround. Youwouldneedto
travelfasterthanlightinordertoendupwhereyoustartedbeforethe
universecametoanend—andthatisnotallowed!
InthefirstkindofFriedmannmodel,whichexpandsandrecollapses,
space is bent in on itself, like the surface of the earth. It is therefore
finite in extent. In the second kind of model, which expands forever,
spaceisbenttheotherway,likethesurfaceofasaddle.Sointhiscase
spaceisinfinite.Finally,inthethirdkindofFriedmannmodel,withjust
thecriticalrateofexpansion,spaceisflat(andthereforeisalsoinfinite).
ButwhichFriedmannmodeldescribesouruniverse?Willtheuniverse
eventually stop expanding and start contracting, or will it expand
forever? To answer this question we need to know the present rate of
expansionoftheuniverseanditspresentaveragedensity.Ifthedensity
islessthanacertaincriticalvalue,determinedbytherateofexpansion,
thegravitationalattractionwillbetooweaktohalttheexpansion.Ifthe
densityisgreaterthanthecriticalvalue,gravitywillstoptheexpansion
atsometimeinthefutureandcausetheuniversetorecollapse.
We can determine the present rate of expansion by measuring the
velocitiesat which other galaxies are movingaway from us, using the
Dopplereffect.Thiscanbedoneveryaccurately.However,thedistances
tothe galaxies arenot very wellknown because wecan only measure
them indirectly. So all we know is that the universe is expanding by
between 5 percent and 10 percent every thousand million years.
However, our uncertainty about the present average density of the
universeisevengreater.Ifweaddupthemassesofallthestarsthatwe
can see in our galaxy and other galaxies, the total is less than one
hundredthoftheamountrequiredtohalttheexpansionoftheuniverse,
even for the lowest estimate of the rate of expansion. Our galaxy and
othergalaxies,however,mustcontainalargeamountof“darkmatter”
thatwecannotseedirectly,butwhichweknowmustbetherebecause
oftheinfluenceofitsgravitationalattractionontheorbitsofstarsinthe
galaxies. Moreover, most galaxies are found in clusters, and we can
similarly infer the presence of yet more dark matter in between the
galaxies in these clusters by its effect on the motion of the galaxies.
Whenweaddupallthisdarkmatter,westillgetonlyaboutonetenthof
theamountrequiredtohalttheexpansion.However,wecannotexclude
thepossibilitythattheremightbesomeotherformofmatter,distributed
almostuniformlythroughouttheuniverse,thatwehavenotyetdetected
andthatmightstillraisetheaveragedensityoftheuniverseuptothe
critical value needed to halt the expansion. The present evidence
thereforesuggeststhattheuniversewillprobablyexpandforever,butall
we can really be sure of is that even if the universe is going to
recollapse,itwon’tdosoforatleastanothertenthousandmillionyears,
since it has already been expanding for at least that long. This should
notundulyworryus:bythattime,unlesswehavecolonizedbeyondthe
SolarSystem,mankindwilllongsincehavediedout,extinguishedalong
withoursun!
AlloftheFriedmannsolutionshavethefeaturethatatsometimein
the past (between ten and twenty thousand million years ago) the
distance between neighboring galaxies must have been zero. At that
time, which we call the big bang, the density of the universe and the
curvatureofspace-timewouldhavebeeninfinite.Becausemathematics
cannotreallyhandleinfinitenumbers,thismeansthatthegeneraltheory
of relativity (on which Friedmann’s solutions are based) predicts that
there is a point in the universe where the theory itself breaks down.
Suchapointisanexampleofwhatmathematicianscallasingularity.In
fact, all our theories of science are formulated on the assumption that
space-timeissmoothandnearlyflat,sotheybreakdownatthebigbang
singularity, where the curvature of space-time is infinite. This means
that even if there were events before the big bang, one could not use
themtodeterminewhatwouldhappenafterward,becausepredictability
wouldbreakdownatthebigbang.
Correspondingly,if,asisthecase,weknowonlywhathashappened
sincethebigbang,wecouldnotdeterminewhathappenedbeforehand.
As far as we are concerned, events before the big bang can have no
consequences,sotheyshouldnotformpartofascientificmodelofthe
universe.We shouldthereforecut them outof the modelandsay that
timehadabeginningatthebigbang.
Manypeopledonotliketheideathattimehasabeginning,probably
becauseitsmacksofdivineintervention.(TheCatholicChurch,onthe
other hand, seized on the big bang model and in 1951 officially
pronouncedittobeinaccordancewiththeBible.)Thereweretherefore
anumberofattemptstoavoidtheconclusionthattherehadbeenabig
bang. The proposal that gained widest support was called the steady
state theory. It was suggested in 1948 by two refugees from Nazi-
occupied Austria, Hermann Bondi and Thomas Gold, together with a
Briton,FredHoyle,whohadworkedwiththemonthedevelopmentof
radar during the war. The idea was that as the galaxies moved away
fromeachother,newgalaxieswerecontinuallyforminginthe gapsin
between, from new matter that was being continually created. The
universewouldthereforelookroughlythesameatalltimesaswellasat
all points of space. The steady state theory required a modification of
generalrelativitytoallowforthecontinualcreationofmatter,butthe
rate that was involved was so low (about one particle per cubic
kilometer per year) that it was not in conflict with experiment. The
theorywasagoodscientifictheory,inthesensedescribedinChapter1:
it was simple and it made definite predictions that could be tested by
observation.Oneofthesepredictionswasthatthenumberofgalaxiesor
similar objects in any given volume of space should be the same
whereverandwheneverwelookintheuniverse.Inthelate1950sand
early 1960s a survey of sources of radio waves from outer space was
carriedoutatCambridgebyagroupofastronomersledbyMartinRyle
(whohadalsoworkedwithBondi,Gold,andHoyleonradarduringthe
war). The Cambridge group showed that most of these radio sources
must lie outside our galaxy (indeed many of them could be identified
withothergalaxies)andalsothatthereweremanymoreweaksources
thanstrongones.Theyinterpretedtheweaksourcesasbeingthemore
distantones,andthestrongeronesasbeingnearer.Thenthereappeared
to be less common sources per unit volume of space for the nearby
sources than for the distant ones. This could mean that we are at the
centerofagreatregionintheuniverseinwhichthesourcesarefewer
thanelsewhere.Alternatively,itcouldmeanthatthesourcesweremore
numerous in the past, at the time that the radio waves left on their
journey to us, than they are now. Either explanation contradicted the
predictions of the steady state theory. Moreover, the discovery of the
microwaveradiationbyPenziasandWilsonin1965alsoindicatedthat
theuniversemusthavebeenmuchdenserinthepast.Thesteadystate
theorythereforehadtobeabandoned.
Anotherattempttoavoidtheconclusionthattheremusthavebeena
bigbang,andthereforeabeginningoftime,wasmadebytwoRussian
scientists, Evgenii Lifshitz and Isaac Khalatnikov, in 1963. They
suggested that the big bang might be a peculiarity of Friedmann’s
models alone, which after all were only approximations to the real
universe. Perhaps, of all the models that were roughly like the real
universe, only Friedmann’s would contain a big bang singularity. In
Friedmann’smodels,thegalaxiesareallmovingdirectlyawayfromeach
other—soitisnotsurprisingthatatsometimeinthepasttheywereall
atthesameplace.Intherealuniverse,however,thegalaxiesarenotjust
movingdirectly awayfromeach other—they alsohave small sideways
velocities.Soinrealitytheyneedneverhavebeenallatexactlythesame
place, only very close together. Perhaps then the current expanding
universe resulted not from a big bang singularity, but from an earlier
contractingphase;astheuniversehadcollapsedtheparticlesinitmight
nothaveallcollided,buthadflownpastandthenawayfromeachother,
producingthepresentexpansionoftheuniverse.Howthencouldwetell
whethertherealuniverseshouldhavestartedoutwithabigbang?What
Lifshitz and Khalatnikov did was to study models of the universe that
were roughly like Friedmann’s models but took account of the
irregularitiesandrandomvelocitiesofgalaxiesintherealuniverse.They
showedthatsuchmodelscouldstartwithabigbang,eventhoughthe
galaxieswerenolongeralwaysmovingdirectlyawayfromeachother,
buttheyclaimedthatthiswasstillonlypossibleincertainexceptional
modelsinwhichthegalaxieswereallmovinginjusttherightway.They
argued that since there seemed to be infinitely more Friedmann-like
models without a big bang singularity than there were with one, we
shouldconcludethattherehadnotinrealitybeenabigbang.Theylater
realized, however, that there was a much more general class of
Friedmann-like models that did have singularities, and in which the
galaxies did not have to be moving any special way. They therefore
withdrewtheirclaimin1970.
TheworkofLifshitzandKhalatnikovwasvaluablebecauseitshowed
thattheuniversecouldhavehadasingularity,abigbang,ifthegeneral
theoryofrelativitywascorrect.However,itdidnotresolvethecrucial
question: Does general relativity predict that our universe should have
hadabigbang,abeginningoftime?Theanswertothiscameoutofa
completely different approach introduced by a British mathematician
andphysicist,RogerPenrose,in1965.Usingthewaylightconesbehave
in general relativity, together with the fact that gravity is always
attractive, he showed that a star collapsing under its own gravity is
trappedinaregionwhosesurfaceeventuallyshrinkstozerosize.And,
sincethesurfaceoftheregionshrinkstozero,sotoomustitsvolume.
All the matter in the star will be compressed into a region of zero
volume,sothedensityofmatterandthecurvatureofspace-timebecome
infinite.Inotherwords,onehasasingularitycontainedwithinaregion
ofspace-timeknownasablackhole.
At first sight, Penrose’s result applied only to stars; it didn’t have
anythingtosayaboutthequestionofwhethertheentireuniversehada
big bang singularity in its past. However, at the time that Penrose
producedhistheorem,Iwasaresearchstudentdesperatelylookingfora
problem with which to complete my Ph.D. thesis. Two years before, I
had been diagnosed as suffering from ALS, commonly known as Lou
Gehrig’sdisease,ormotorneurondisease,andgiventounderstandthatI
hadonlyoneortwomoreyearstolive.Inthesecircumstancestherehad
not seemed much point in working on my Ph.D.—I did not expect to
survivethatlong.YettwoyearshadgonebyandIwasnotthatmuch
worse. In fact, things were going rather well for me and I had gotten
engagedtoaverynice girl,JaneWilde.Butinordertoget married,I
neededajob,andinordertogetajob,IneededaPh.D.
In 1965 I read about Penrose’s theorem that any body undergoing
gravitationalcollapsemusteventuallyformasingularity.Isoonrealized
thatifonereversedthedirectionoftimeinPenrose’stheorem,sothat
thecollapsebecameanexpansion,theconditionsofhistheoremwould
stillhold, provided the universewere roughly like aFriedmannmodel
on large scales at the present time. Penrose’s theorem had shown that
anycollapsingstarmustendinasingularity;thetime-reversedargument
showed that any Friedmann-like expanding universe must have begun
withasingularity.Fortechnicalreasons,Penrose’stheoremrequiredthat
theuniversebeinfiniteinspace.SoIcouldinfactuseittoprovethat
there should be a singularity only if the universe was expanding fast
enough to avoid collapsing again (since only those Friedmann models
wereinfiniteinspace).
DuringthenextfewyearsIdevelopednewmathematicaltechniques
to remove this and other technical conditions from the theorems that
provedthatsingularitiesmustoccur.Thefinalresultwasajointpaper
by Penrose and myself in 1970, which at last proved that there must
havebeenabigbangsingularityprovidedonlythatgeneralrelativityis
correctandtheuniversecontainsasmuchmatterasweobserve.There
wasalotofoppositiontoourwork,partlyfromtheRussiansbecauseof
their Marxist belief in scientific determinism, and partly from people
whofeltthatthewholeideaofsingularitieswasrepugnantandspoiled
thebeautyofEinstein’stheory.However,onecannotreallyarguewitha
mathematical theorem. So in the end our work became generally
accepted and nowadays nearly everyone assumes that the universe
started with a big bang singularity. It is perhaps ironic that, having
changed my mind, I am now trying to convince other physicists that
therewasinfactnosingularityatthebeginningoftheuniverse—aswe
shall see later, it can disappear once quantum effects are taken into
account.
Wehaveseeninthischapterhow,inlessthanhalfacentury,man’s
view of the universe, formed over millennia, has been transformed.
Hubble’sdiscoverythattheuniversewasexpanding,andtherealization
oftheinsignificanceofourownplanetinthevastnessoftheuniverse,
were just the starting point. As experimental and theoretical evidence
mounted, it became more and more clear that the universe must have
hadabeginningintime,untilin1970thiswasfinallyprovedbyPenrose
andmyself,onthebasisofEinstein’sgeneraltheoryof relativity.That
proof showed that general relativity is only an incomplete theory: it
cannottellus howthe universestarted off,because itpredicts thatall
physical theories, including itself, break down at the beginning of the
universe.However,generalrelativityclaimstobeonlyapartialtheory,
so what the singularity theorems really show is that there must have
beenatimeintheveryearlyuniversewhentheuniversewassosmall
thatonecouldnolongerignorethesmall-scaleeffectsoftheothergreat
partialtheoryofthetwentiethcentury,quantummechanics.Atthestart
of the 1970s, then, we were forced to turn our search for an
understandingoftheuniversefromourtheoryoftheextraordinarilyvast
to our theory of the extraordinarily tiny. That theory, quantum
mechanics, will be described next, before we turn to the efforts to
combinethetwopartialtheoriesintoasinglequantumtheoryofgravity.
T
CHAPTER4
THE
UNCERTAINTY
PRINCIPLE
he success of scientific theories, particularly Newton’s theory of
gravity, led the French scientist the Marquis de Laplace at the
beginning of the nineteenth century to argue that the universe was
completelydeterministic.Laplacesuggestedthatthereshouldbeasetof
scientific laws that would allow us to predict everything that would
happen in the universe, if only we knew the complete state of the
universeatonetime.Forexample,ifweknewthepositionsandspeeds
ofthesunandtheplanetsatonetime,thenwecoulduseNewton’slaws
tocalculatethestateoftheSolarSystematanyothertime.Determinism
seemsfairlyobviousinthiscase,butLaplacewentfurthertoassumethat
there were similar laws governing everything else, including human
behavior.
Thedoctrineofscientificdeterminismwasstronglyresistedbymany
people, who felt that it infringed God’s freedom to intervene in the
world,butitremainedthestandardassumptionofscienceuntiltheearly
yearsofthiscentury.Oneofthefirstindicationsthatthisbeliefwould
havetobeabandonedcame whencalculationsbythe Britishscientists
LordRayleighandSirJamesJeanssuggestedthatahotobject,orbody,
suchasastar,mustradiateenergyataninfiniterate.Accordingtothe
laws we believed at the time, a hot body ought to give off
electromagnetic waves (such as radio waves, visible light, or X rays)
equally at all frequencies. For example, a hot body should radiate the
sameamountofenergyinwaveswithfrequenciesbetweenoneandtwo
million million waves a second as in waves with frequencies between
twoandthreemillionmillionwavesasecond.Nowsincethenumberof
waves a second is unlimited, this would mean that the total energy
radiatedwouldbeinfinite.
Inordertoavoidthisobviouslyridiculousresult,theGermanscientist
MaxPlancksuggestedin1900thatlight,Xrays,andotherwavescould
notbeemittedatanarbitraryrate,butonlyincertainpacketsthathe
calledquanta.Moreover,eachquantumhadacertainamountofenergy
that was greater the higher the frequency of the waves, so at a high
enoughfrequencytheemissionofasinglequantumwouldrequiremore
energythanwasavailable.Thustheradiationathighfrequencieswould
be reduced, and so the rate at which the body lost energy would be
finite.
The quantum hypothesis explained the observed rate of emission of
radiationfromhotbodiesverywell,butitsimplicationsfordeterminism
were not realized until 1926, when another German scientist, Werner
Heisenberg, formulated his famous uncertainty principle. In order to
predictthefuturepositionandvelocityofaparticle,onehastobeable
tomeasureitspresentpositionandvelocityaccurately.Theobviousway
todothisistoshinelightontheparticle.Someofthewavesoflightwill
bescatteredbytheparticleandthiswillindicateitsposition.However,
one will not be able to determine the position of the particle more
accurately than the distance between the wave crests of light, so one
needstouselightofashortwavelengthinordertomeasuretheposition
of the particle precisely. Now, by Planck’s quantum hypothesis, one
cannotuseanarbitrarilysmallamountoflight;onehastouseatleast
one quantum. This quantum will disturb the particle and change its
velocity in a way that cannot be predicted. Moreover, the more
accuratelyonemeasurestheposition,theshorterthewavelengthofthe
light that one needs and hence the higher the energy of a single
quantum. So the velocity of the particle will be disturbed by a larger
amount. In other words, the more accurately you try to measure the
position of the particle, the less accurately you can measure its speed,
andviceversa.Heisenbergshowedthattheuncertaintyinthepositionof
the particle times the uncertainty in its velocity times the mass of the
particlecanneverbesmallerthanacertainquantity,whichisknownas
Planck’s constant. Moreover, this limit does not depend on the way in
whichonetriestomeasurethepositionorvelocityoftheparticle,oron
thetypeofparticle:Heisenberg’suncertaintyprincipleisafundamental,
inescapablepropertyoftheworld.
The uncertainty principle had profound implications for the way in
whichweviewtheworld.Evenaftermorethanseventyyearstheyhave
not been fully appreciated by many philosophers, and are still the
subjectofmuchcontroversy.Theuncertaintyprinciplesignaledanend
toLaplace’sdreamofatheoryofscience,amodeloftheuniversethat
wouldbe completely deterministic: one certainly cannot predict future
events exactly if one cannot even measure the present state of the
universeprecisely!Wecouldstillimaginethatthereisasetoflawsthat
determine events completely for some supernatural being, who could
observethepresentstateoftheuniversewithoutdisturbingit.However,
such models of the universe are not of much interest to us ordinary
mortals.It seems better to employ the principleof economy known as
Occam’srazorandcutoutallthefeaturesofthetheorythatcannotbe
observed. This approach led Heisenberg, Erwin Schrödinger, and Paul
Dirac in the 1920s to reformulate mechanics into a new theory called
quantum mechanics, based on the uncertainty principle. In this theory
particles no longer had separate, well-defined positions and velocities
that could not be observed. Instead, they had a quantum state, which
wasacombinationofpositionandvelocity.
Ingeneral,quantummechanicsdoesnotpredictasingledefiniteresult
for an observation. Instead, it predicts a number of different possible
outcomesandtellsushowlikelyeachoftheseis.Thatistosay,ifone
madethesamemeasurementonalargenumberofsimilarsystems,each
ofwhichstartedoffinthesameway,onewouldfindthattheresultof
the measurement would be A in a certain number of cases, B in a
differentnumber,andsoon.Onecouldpredicttheapproximatenumber
oftimesthattheresultwouldbeAorB,butonecouldnotpredictthe
specific result of an individual measurement. Quantum mechanics
therefore introduces an unavoidable element of unpredictability or
randomnessintoscience.Einsteinobjectedtothisverystrongly,despite
the important role he had played in the development of these ideas.
Einstein was awarded the Nobel Prize for his contribution to quantum
theory. Nevertheless, Einstein never accepted that the universe was
governed by chance; his feelings were summed up in his famous
statement“Goddoesnotplaydice.”Mostotherscientists,however,were
willing to accept quantum mechanics because it agreed perfectly with
experiment.Indeed,ithasbeenanoutstandinglysuccessfultheoryand
underlies nearly all of modern science and technology. It governs the
behavior of transistors and integrated circuits, which are the essential
componentsofelectronicdevicessuchastelevisionsandcomputers,and
is also the basis of modern chemistry and biology. The only areas of
physical science into which quantum mechanics has not yet been
properly incorporated are gravity and the large-scale structure of the
universe.
Althoughlightismadeupofwaves,Planck’squantumhypothesistells
usthatin someways itbehaves asif itwere composedofparticles: it
can be emitted or absorbed only in packets, or quanta. Equally,
Heisenberg’suncertaintyprincipleimpliesthatparticlesbehaveinsome
respects like waves: they do not have a definite position but are
“smeared out” with a certain probability distribution. The theory of
quantum mechanics is based on an entirely new type of mathematics
thatnolongerdescribestherealworldintermsofparticlesandwaves;it
is only the observations of the world that may be described in those
terms.Thereisthusadualitybetweenwavesandparticlesinquantum
mechanics:forsomepurposesitishelpfultothinkofparticlesaswaves
and for other purposes it is better to think of waves as particles. An
important consequence of this is that one can observe what is called
interferencebetweentwosetsofwavesorparticles.Thatistosay,the
crestsofonesetofwavesmaycoincidewiththetroughsoftheotherset.
Thetwosetsofwavesthencanceleachotheroutratherthanaddingup
toastrongerwaveasonemightexpect(Fig.4.1).Afamiliarexampleof
interferenceinthecaseoflightisthecolorsthatareoftenseeninsoap
bubbles.Thesearecausedbyreflectionoflightfromthetwosidesofthe
thinfilmofwaterformingthebubble.Whitelightconsistsoflightwaves
ofalldifferentwavelengths,orcolors.Forcertainwavelengthsthecrests
ofthewavesreflectedfromonesideofthesoapfilmcoincidewiththe
troughsreflectedfromtheotherside.Thecolorscorrespondingtothese
wavelengthsareabsentfromthereflectedlight,whichthereforeappears
tobecolored.
Interference can also occur for particles, because of the duality
introduced by quantum mechanics. A famous example is the so-called
two-slit experiment (Fig. 4.2). Consider a partition with two narrow
parallel slits in it. On one side of the partition one places a source of
lightofaparticularcolor(thatis,ofaparticularwavelength).Mostof
thelightwillhitthepartition,butasmallamountwillgothroughthe
slits. Now suppose one places a screen on the far side of the partition
fromthelight.Anypointonthescreenwillreceivewavesfromthetwo
slits.However,ingeneral,thedistancethelighthastotravelfromthe
source to the screen via the two slits will be different.This will mean
thatthewavesfromtheslitswillnotbeinphasewitheachotherwhen
they arrive at the screen: in some places the waves will cancel each
otherout, and in othersthey will reinforce eachother.The result is a
characteristicpatternoflightanddarkfringes.
FIGURE4.1
FIGURE4.2
Theremarkablethingisthatonegetsexactlythesamekindoffringes
if one replaces the source of light by a source of particles such as
electronswithadefinitespeed(thismeansthatthecorrespondingwaves
haveadefinitelength).Itseemsthemorepeculiarbecauseifoneonly
hasoneslit,onedoesnotgetanyfringes,justauniformdistributionof
electrons across the screen. One might therefore think that opening
another slit would just increase the number of electrons hitting each
pointofthescreen,but,becauseofinterference,itactuallydecreasesit
insomeplaces.Ifelectronsaresentthroughtheslitsoneatatime,one
wouldexpecteachtopassthroughoneslitortheother,andsobehave
just as if the slit it passed through were the only one there—giving a
uniformdistributiononthescreen.Inreality,however,evenwhenthe
electronsaresentoneatatime,thefringesstillappear.Eachelectron,
therefore,mustbepassingthroughbothslitsatthesametime!
Thephenomenonofinterferencebetweenparticleshasbeencrucialto
ourunderstandingofthestructureofatoms,thebasicunitsofchemistry
and biology and the building blocks out of which we, and everything
aroundus,aremade.Atthebeginningofthiscenturyitwasthoughtthat
atoms were rather like the planets orbiting the sun, with electrons
(particles of negative electricity) orbiting around a central nucleus,
which carried positive electricity. The attraction between the positive
and negative electricity was supposed to keep the electrons in their
orbitsinthesamewaythatthegravitationalattractionbetweenthesun
andtheplanetskeepsthe planetsintheirorbits.Thetroublewiththis
was that the laws of mechanics and electricity, before quantum
mechanics,predictedthattheelectronswouldloseenergyandsospiral
inwarduntiltheycollidedwiththenucleus.Thiswouldmeanthatthe
atom,and indeedallmatter, should rapidlycollapse to astate of very
highdensity.ApartialsolutiontothisproblemwasfoundbytheDanish
scientistNielsBohrin1913.Hesuggestedthatmaybetheelectronswere
notabletoorbitatjustanydistancefromthecentralnucleusbutonlyat
certain specified distances. If one also supposed that only one or two
electronscouldorbitatanyoneofthesedistances,thiswouldsolvethe
problem of the collapse of the atom, because the electrons could not
spiralinanyfartherthantofilluptheorbitswiththeleastdistancesand
energies.
This model explained quite well the structure of the simplest atom,
hydrogen,whichhasonlyoneelectronorbitingaroundthenucleus.But
itwasnotclearhowoneoughttoextendittomorecomplicatedatoms.
Moreover, the idea of a limited set of allowed orbits seemed very
arbitrary.Thenewtheoryofquantummechanicsresolvedthisdifficulty.
Itrevealedthatanelectronorbitingaroundthenucleuscouldbethought
of as a wave, with a wavelength that depended on its velocity. For
certain orbits, the length of the orbit would correspond to a whole
number (as opposed to a fractional number) of wavelengths of the
electron.Fortheseorbitsthewavecrestwouldbeinthesameposition
each time round, so the waves would add up: these orbits would
correspondtoBohr’sallowedorbits.However,fororbitswhoselengths
were not a whole number of wavelengths, each wave crest would
eventually be canceled out by a trough as the electrons went round;
theseorbitswouldnotbeallowed.
A nice way of visualizing the wave/particle duality is the so-called
sum over histories introduced by the American scientist Richard
Feynman.Inthisapproachtheparticleisnotsupposedtohaveasingle
history or path in space-time, as it would in a classical, nonquantum
theory.InsteaditissupposedtogofromAtoBbyeverypossiblepath.
Witheachpaththereareassociatedacoupleofnumbers:onerepresents
thesizeofawaveandtheotherrepresentsthepositioninthecycle(i.e.,
whetheritisatacrestoratrough).TheprobabilityofgoingfromAtoB
is found by adding up the waves for all the paths. In general, if one
comparesasetofneighboringpaths,thephasesorpositionsinthecycle
willdiffergreatly.Thismeansthatthewavesassociatedwiththesepaths
will almost exactly cancel each other out. However, for some sets of
neighboring paths the phase will not vary much between paths. The
waves for these paths will not cancel out. Such paths correspond to
Bohr’sallowedorbits.
With these ideas, in concrete mathematical form, it was relatively
straightforward to calculate the allowed orbits in more complicated
atomsandeveninmolecules,whicharemadeupofanumberofatoms
held together by electrons in orbits that go round more than one
nucleus.Sincethestructureofmoleculesandtheirreactionswitheach
otherunderlieallofchemistryandbiology,quantummechanicsallows
usinprincipletopredictnearlyeverythingweseearoundus,withinthe
limits set by the uncertainty principle. (In practice, however, the
calculationsrequiredforsystemscontainingmorethanafew electrons
aresocomplicatedthatwecannotdothem.)
Einstein’sgeneraltheoryofrelativityseemstogovernthelarge-scale
structureoftheuniverse.Itiswhatiscalledaclassicaltheory;thatis,it
does not take account of the uncertainty principle of quantum
mechanics,asitshouldforconsistencywithothertheories.Thereason
thatthisdoesnotleadtoanydiscrepancywithobservationisthatallthe
gravitational fields that we normally experience are very weak.
However, the singularity theorems discussed earlier indicate that the
gravitationalfieldshouldgetverystronginatleasttwosituations,black
holes and the big bang. In such strong fields the effects of quantum
mechanics should be important. Thus, in a sense, classical general
relativity, by predicting points of infinite density, predicts its own
downfall,justasclassical(thatis,nonquantum)mechanicspredictedits
downfallbysuggestingthatatomsshouldcollapsetoinfinitedensity.We
do not yet have a complete consistent theory that unifies general
relativity and quantum mechanics, but we do know a number of the
features it should have. The consequences that these would have for
blackholesandthebigbangwillbedescribedinlaterchapters.Forthe
moment,however,weshallturntotherecentattemptstobringtogether
our understanding of the other forces of nature into a single, unified
quantumtheory.
A
CHAPTER5
ELEMENTARY
PARTICLESAND
THEFORCESOF
NATURE
ristotlebelievedthatallthematterintheuniversewasmadeupof
fourbasicelements—earth,air,fire,andwater.Theseelementswere
actedonbytwoforces:gravity,thetendencyforearthandwatertosink,
and levity, the tendency for air and fire to rise. This division of the
contentsoftheuniverseintomatterandforcesisstillusedtoday.
Aristotlebelievedthatmatterwascontinuous,thatis,onecoulddivide
a piece of matter into smaller and smaller bits without any limit: one
never came up against a grain of matter that could not be divided
further. A few Greeks, however, such as Democritus, held that matter
was inherently grainy and that everything was made up of large
numbers of various different kinds of atoms. (The word atom means
“indivisible” in Greek.) For centuries the argument continued without
any real evidence on either side, but in 1803 the British chemist and
physicistJohnDaltonpointedoutthatthefactthatchemicalcompounds
always combined in certain proportions could be explained by the
groupingtogetherofatomstoformunitscalledmolecules.However,the
argumentbetweenthetwoschoolsofthoughtwasnotfinallysettledin
favor of the atomists until the early years of this century. One of the
important pieces of physical evidence was provided by Einstein. In a
paperwrittenin1905,afewweeksbeforethefamouspaperonspecial
relativity,EinsteinpointedoutthatwhatwascalledBrownianmotion—
theirregular, randommotion ofsmallparticles ofdust suspendedin a
liquid—couldbeexplainedastheeffectofatomsoftheliquidcolliding
withthedustparticles.
Bythistimetherewerealreadysuspicionsthattheseatomswerenot,
afterall,indivisible.SeveralyearspreviouslyafellowofTrinityCollege,
Cambridge,J.J.Thomson,haddemonstratedtheexistenceofaparticle
ofmatter,calledtheelectron,thathadamasslessthanonethousandth
of that of the lightest atom. He used a setup rather like a modern TV
picturetube:ared-hotmetalfilamentgaveofftheelectrons,andbecause
thesehaveanegativeelectriccharge,anelectricfieldcouldbeusedto
accelerate them toward a phosphor-coated screen. When they hit the
screen,flashesoflightweregenerated.Soonitwas realizedthatthese
electrons must be coming from within the atoms themselves, and in
1911 the New Zealand physicist Ernest Rutherford finally showed that
theatomsofmatterdohaveinternalstructure:theyaremadeupofan
extremely tiny, positively charged nucleus, around which a number of
electronsorbit.Hededucedthisby analyzingtheway inwhichalpha-
particles,whicharepositivelychargedparticlesgivenoffbyradioactive
atoms,aredeflectedwhentheycollidewithatoms.
Atfirstitwasthoughtthatthenucleusoftheatomwasmadeupof
electrons and different numbers of a positively charged particle called
the proton, from the Greek word meaning “first,” because it was
believed to be the fundamental unit from which matter was made.
However, in 1932 a colleague of Rutherford’s at Cambridge, James
Chadwick,discoveredthatthenucleuscontainedanotherparticle,called
the neutron, which had almost the same mass as a proton but no
electricalcharge. Chadwickreceived the NobelPrizefor his discovery,
andwaselectedMasterofGonvilleandCaiusCollege,Cambridge(the
collegeofwhichIamnowafellow).HelaterresignedasMasterbecause
ofdisagreementswiththeFellows.Therehadbeenabitterdisputeinthe
collegeeversinceagroupofyoungFellowsreturningafterthewarhad
votedmanyoftheoldFellowsoutofthecollegeofficestheyhadheldfor
alongtime.Thiswasbeforemytime;Ijoinedthecollegein1965atthe
tail end of the bitterness, when similar disagreements forced another
NobelPrize—winningMaster,SirNevillMott,toresign.
Uptoaboutthirtyyearsago,itwasthoughtthatprotonsandneutrons
were “elementary” particles, but experiments in which protons were
collided with other protons or electrons at high speeds indicated that
they were in fact made up of smaller particles. These particles were
namedquarksbytheCaltechphysicistMurrayGell-Mann,whowonthe
NobelPrizein1969forhisworkonthem.Theoriginofthenameisan
enigmaticquotationfromJamesJoyce:“ThreequarksforMusterMark!”
Thewordquarkissupposedtobepronouncedlikequart,butwithakat
theendinsteadofat,butisusuallypronouncedtorhymewithlark.
There are a number of different varieties of quarks: there are six
“flavors,”whichwecallup,down,strange,charmed,bottom,andtop.
Thefirstthreeflavorshadbeenknownsincethe1960sbutthecharmed
quarkwasdiscoveredonlyin1974,thebottomin1977,andthetopin
1995. Each flavor comes in three “colors,” red, green, and blue. (It
shouldbeemphasizedthatthesetermsarejustlabels:quarksaremuch
smallerthanthewavelengthofvisiblelightandsodonothaveanycolor
inthenormalsense.Itisjustthatmodernphysicistsseemtohavemore
imaginative ways of naming new particles and phenomena—they no
longerrestrictthemselvestoGreek!)Aprotonorneutronismadeupof
three quarks, one of each color. A proton contains two up quarks and
one down quark; a neutron contains two down and one up. We can
createparticlesmadeupoftheotherquarks(strange,charmed,bottom,
andtop),buttheseallhaveamuchgreatermassanddecayveryrapidly
intoprotonsandneutrons.
We now know that neither the atoms nor the protons and neutrons
within them are indivisible. So the question is: what are the truly
elementaryparticles,thebasicbuildingblocksfromwhicheverythingis
made?Sincethewavelengthoflightismuchlargerthanthesizeofan
atom,wecannothopeto“look”atthepartsofanatomintheordinary
way.Weneedtousesomethingwithamuchsmallerwavelength.Aswe
sawinthelastchapter,quantummechanicstellsusthatallparticlesare
infactwaves,andthatthehighertheenergyofaparticle,thesmaller
thewavelengthof thecorrespondingwave.Sothebest answerwecan
givetoourquestiondependsonhowhighaparticleenergywehaveat
ourdisposal,becausethisdeterminesonhowsmallalengthscalewecan
look. These particle energies are usually measured in units called
electronvolts.(InThomson’sexperimentswithelectrons,wesawthathe
used an electric field to accelerate the electrons. The energy that an
electrongainsfromanelectricfieldofonevoltiswhatisknownasan
electronvolt.)Inthenineteenthcentury,whentheonlyparticleenergies
that people knew how to use were the low energies of a few electron
volts generated by chemical reactions such as burning, it was thought
thatatomswerethesmallestunit.InRutherford’sexperiment,thealpha-
particles had energies of millions of electron volts. More recently, we
havelearnedhowtouseelectromagneticfieldstogiveparticlesenergies
ofatfirstmillionsandthenthousandsofmillionsofelectronvolts.And
sowe knowthat particles thatwere thoughttobe “elementary”thirty
yearsagoare,infact,madeupofsmallerparticles.Maythese,aswego
tostill higherenergies, inturn befound to bemade fromstill smaller
particles? This is certainly possible, but we do have some theoretical
reasonsforbelievingthatwehave,orareverynearto,aknowledgeof
theultimatebuildingblocksofnature.
Using the wave/particle duality discussed in the last chapter,
everythingintheuniverse,includinglightandgravity,canbedescribed
in terms of particles. These particles have a property called spin. One
wayofthinkingofspinistoimaginetheparticlesaslittletopsspinning
about an axis. However, this can be misleading, because quantum
mechanicstellsusthattheparticlesdonothaveanywell-definedaxis.
Whatthespinofaparticlereallytellsusiswhattheparticlelookslike
fromdifferentdirections.Aparticleofspin0islikeadot:itlooksthe
samefromeverydirection(Fig.5.1-i).Ontheotherhand,aparticleof
spin1islikeanarrow:itlooksdifferentfromdifferentdirections(Fig.
5.1-ii).Only if one turns it roundacomplete revolution (360 degrees)
does the particle look the same. A particle of spin 2 is like a double-
headedarrow(Fig.5.1-iii):itlooksthesameifoneturnsitroundhalfa
revolution(180degrees).Similarly,higherspinparticleslookthesameif
oneturnsthemthrough smallerfractionsofa completerevolution.All
this seems fairly straightforward, but the remarkable fact is that there
are particles that do not look the same if one turnsthem through just
one revolution: you have to turn them through two complete
revolutions!Suchparticlesaresaidtohavespin½.
All the known particles in the universe can be divided into two
groups:particlesofspin½,whichmakeupthematterintheuniverse,
and particles of spin 0, 1, and 2, which, as we shall see, give rise to
forces between the matter particles. The matter particles obey what is
called Pauli’s exclusion principle. This was discovered in 1925 by an
Austrian physicist, Wolfgang Pauli—for which he received the Nobel
Prizein1945.Hewasthearchetypaltheoreticalphysicist:itwassaidof
himthatevenhispresenceinthesametownwouldmakeexperiments
go wrong! Pauli’s exclusion principle says that two similar particles
cannotexistinthesamestate;thatis,theycannothaveboththesame
position and the same velocity, within the limits given by the
uncertainty principle. The exclusion principle is crucial because it
explains why matter particles do not collapse to a state of very high
density under the influence of the forces produced by the particles of
spin 0, 1, and 2: if the matter particles have very nearly the same
positions,theymusthavedifferentvelocities,whichmeansthattheywill
not stay in the same position for long. If the world had been created
withoutthe exclusion principle,quarks wouldnotform separate, well-
definedprotonsandneutrons.Norwouldthese,togetherwithelectrons,
form separate, well-defined atoms. They would all collapse to form a
roughlyuniform,dense“soup.”
FIGURE5.1
Aproperunderstandingoftheelectronandotherspin-½particlesdid
notcomeuntil1928,whenatheorywasproposedbyPaulDirac,who
later was elected to the Lucasian Professorship of Mathematics at
Cambridge(thesameprofessorshipthatNewtonhadonceheldandthat
Inowhold).Dirac’stheorywasthefirstofitskindthatwasconsistent
with both quantum mechanics and the special theory of relativity. It
explainedmathematicallywhytheelectronhadspin½;thatis,whyit
didn’t look the same if you turned it through only one complete
revolution, but did if you turned it through two revolutions. It also
predicted that the electron should have a partner: an antielectron, or
positron.Thediscoveryofthepositronin1932confirmedDirac’stheory
andled to his beingawarded the Nobel Prizefor physics in 1933.We
now know that every particle has an antiparticle, with which it can
annihilate.(Inthe caseoftheforce-carryingparticles,the antiparticles
are the same as the particles themselves.) There could be whole
antiworlds and antipeople made out of antiparticles. However, if you
meetyourantiself,don’tshakehands!Youwouldbothvanishinagreat
flash of light. The question of why there seem to be so many more
particlesthanantiparticlesaroundusisextremelyimportant,andIshall
returntoitlaterinthechapter.
In quantum mechanics, the forces or interactions between matter
particlesareallsupposedtobecarriedbyparticlesofintegerspin—0,1,
or 2. What happens is that a matter particle, such as an electron or a
quark, emits a force-carrying particle. The recoil from this emission
changes the velocity of the matter particle. The force-carrying particle
thencollideswithanothermatterparticleandisabsorbed.Thiscollision
changesthevelocityofthesecondparticle,justasiftherehad beena
forcebetweenthetwomatterparticles.Itisanimportantpropertyofthe
force-carrying particles that they do not obey the exclusion principle.
Thismeansthatthereisnolimittothenumberthatcanbeexchanged,
andsotheycangiverisetoastrongforce.However,iftheforce-carrying
particleshaveahighmass,itwillbedifficulttoproduceandexchange
themoveralargedistance.Sotheforcesthattheycarrywillhaveonlya
shortrange. On the other hand, if theforce-carryingparticles have no
mass of their own, the forces will be long range. The force-carrying
particles exchanged between matter particles are said to be virtual
particles because, unlike “real” particles, they cannot be directly
detectedbya particledetector.Weknowtheyexist,however,because
theydohaveameasurableeffect:theygiverisetoforcesbetweenmatter
particles.Particlesofspin0,1,or2doalsoexistinsomecircumstances
asrealparticles,whentheycanbedirectlydetected.Theythenappear
to us as what a classical physicist would call waves, such as waves of
light or gravitational waves. They may sometimes be emitted when
matter particles interact with each other by exchanging virtual force-
carryingparticles.(Forexample,theelectricrepulsiveforcebetweentwo
electronsisduetotheexchangeofvirtualphotons,whichcanneverbe
directlydetected; but if oneelectron moves past another,real photons
maybegivenoff,whichwedetectaslightwaves.)
Force-carryingparticlescanbegroupedintofourcategoriesaccording
tothestrengthoftheforcethattheycarryandtheparticleswithwhich
theyinteract.Itshouldbeemphasizedthatthisdivisionintofourclasses
isman-made;itisconvenientfortheconstructionofpartialtheories,but
it may not correspond to anything deeper. Ultimately, most physicists
hopetofindaunifiedtheorythatwillexplainallfourforcesasdifferent
aspectsofasingleforce.Indeed,manywouldsaythisistheprimegoal
ofphysicstoday.Recently,successfulattemptshavebeenmadetounify
threeofthefourcategoriesofforce—and Ishall describethese inthis
chapter. The question of the unification of the remaining category,
gravity,weshallleavetilllater.
The first category is the gravitational force. This force is universal,
thatis,everyparticlefeelstheforceofgravity,accordingtoitsmassor
energy.Gravityistheweakestofthefourforcesbyalongway;itisso
weak that we would not notice it at all were it not for two special
properties that it has: it can act over large distances, and it is always
attractive. This means that the very weak gravitational forces between
the individual particles in two large bodies, such as the earth and the
sun,canalladduptoproduceasignificantforce.Theotherthreeforces
are either short range, or are sometimes attractive and sometimes
repulsive,sotheytendtocancelout.Inthequantummechanicalwayof
lookingatthegravitationalfield,theforcebetweentwomatterparticles
ispicturedasbeingcarriedbyaparticleofspin2calledthegraviton.
Thishasnomassofitsown,sotheforcethatitcarriesislongrange.The
gravitational force between the sun and the earth is ascribed to the
exchange of gravitons between the particles that make up these two
bodies. Although the exchanged particles are virtual, they certainlydo
produce a measurable effect—they make the earth orbit the sun! Real
gravitons make up what classical physicists would call gravitational
waves,which are very weak—and so difficultto detect that they have
notyetbeenobserved.
The next category is the electromagnetic force, which interacts with
electrically charged particles like electrons and quarks, but not with
uncharged particles such as gravitons. It is much stronger than the
gravitational force: the electromagnetic force between two electrons is
about a million million million million million million million (1 with
forty-two zeros after it) times bigger than the gravitational force.
However,therearetwokindsofelectriccharge,positiveandnegative.
The force between two positive charges is repulsive, as is the force
between two negative charges, but the force is attractive between a
positiveand a negativecharge. A large body,such as the earth orthe
sun, contains nearly equal numbers of positive and negative charges.
Thustheattractiveandrepulsiveforcesbetweentheindividualparticles
nearlycanceleachotherout,andthereisverylittlenetelectromagnetic
force. However, on the small scales of atoms and molecules,
electromagnetic forces dominate. The electromagnetic attraction
betweennegativelychargedelectronsandpositivelychargedprotonsin
thenucleuscausestheelectronstoorbitthenucleusoftheatom,justas
gravitational attraction causes the earth to orbit the sun. The
electromagneticattractionispicturedasbeing causedbythe exchange
oflargenumbersofvirtualmasslessparticlesofspin1,calledphotons.
Again, the photons that are exchanged are virtual particles. However,
whenanelectronchangesfromoneallowedorbittoanotheronenearer
tothenucleus,energyisreleasedandarealphotonisemitted—which
can be observed as visible light by the human eye, if it has the right
wavelength,orbyaphotondetectorsuchasphotographicfilm.Equally,
ifarealphotoncollideswithanatom,itmaymoveanelectronfroman
orbitnearerthenucleustoonefartheraway.Thisusesuptheenergyof
thephoton,soitisabsorbed.
The third category is called the weak nuclear force, which is
responsible for radioactivity and which acts on all matter particles of
spin ½, but not on particles of spin 0, 1, or 2, such as photons and
gravitons.Theweaknuclearforcewasnotwellunderstooduntil1967,
whenAbdusSalamatImperialCollege,London,andStevenWeinbergat
Harvard both proposed theories that unified this interaction with the
electromagnetic force, just as Maxwell had unified electricity and
magnetism about a hundred years earlier. They suggested that in
additionto the photon, therewerethree other spin-1 particles,known
collectivelyasmassivevectorbosons,thatcarriedtheweakforce.These
werecalledW+(pronouncedWplus),W(pronouncedWminus),andZ°
(pronouncedZnaught),andeachhadamassofaround100GeV(GeV
standsforgigaelectron-volt,oronethousandmillionelectronvolts).The
Weinberg-Salam theory exhibits a property known as spontaneous
symmetry breaking. This means that what appear to be a number of
completelydifferentparticlesatlowenergiesareinfactfoundtobeall
the same type of particle, only in different states. At high energies all
theseparticlesbehavesimilarly.Theeffectisratherlikethebehaviorof
arouletteballonaroulettewheel.Athighenergies(whenthewheelis
spun quickly) the ball behaves in essentially only one way—it rolls
round and round. But as the wheel slows, the energy of the ball
decreases,andeventuallytheballdropsintooneofthethirty-sevenslots
in the wheel. In other words, at low energies there are thirty-seven
differentstatesinwhichtheballcanexist.If,forsomereason,wecould
only observe the ball at low energies, we would then think that there
werethirty-sevendifferenttypesofball!
In the Weinberg-Salam theory, at energies much greater than 100
GeV, the three new particles and the photon would all behave in a
similar manner. But at the lower particle energies that occur in most
normalsituations,thissymmetrybetweentheparticleswouldbebroken.
W+, W, and Z° would acquire large masses, making the forces they
carry have a very short range. At the time that Salam and Weinberg
proposed their theory, few people believed them, and particle
acceleratorswerenotpowerfulenoughtoreachtheenergiesof100GeV
requiredtoproducerealW+,W,orZ°particles.However,overthenext
ten years or so, the other predictions of the theory at lower energies
agreedsowellwithexperimentthat,in1979,SalamandWeinbergwere
awarded the Nobel Prize for physics, together with Sheldon Glashow,
also at Harvard, who had suggested similar unified theories of the
electromagnetic and weak nuclear forces. The Nobel committee was
sparedtheembarrassmentofhavingmadeamistakebythediscoveryin
1983 at CERN (European Centre for Nuclear Research) of the three
massive partners of the photon, with the correct predicted masses and
other properties. Carlo Rubbia, who led the team of several hundred
physicists that made the discovery, received the Nobel Prize in 1984,
alongwithSimonvanderMeer,theCERNengineerwhodevelopedthe
antimatterstoragesystememployed.(Itisverydifficulttomakeamark
inexperimentalphysicsthesedaysunlessyouarealreadyatthetop!)
Thefourthcategoryisthestrongnuclearforce,whichholdsthequarks
togetherintheprotonandneutron,andholdstheprotonsandneutrons
togetherinthenucleusofanatom.Itisbelievedthatthisforceiscarried
by another spin-1 particle, called the gluon, which interacts only with
itself and with the quarks. The strong nuclear force has a curious
property called confinement: it always binds particles together into
combinationsthathavenocolor.Onecannothaveasinglequarkonits
ownbecauseitwouldhaveacolor(red,green,orblue).Instead,ared
quark has to be joined to a green and a blue quark by a “string” of
gluons (red + green + blue = white). Such a triplet constitutes a
protonoraneutron.Anotherpossibilityisapairconsistingofaquark
and an antiquark (red + antired, or green + antigreen, or blue +
antiblue=white).Suchcombinationsmakeuptheparticlesknownas
mesons, which are unstable because the quark and antiquark can
annihilateeachother,producingelectronsandotherparticles.Similarly,
confinement prevents one having a single gluon on its own, because
gluonsalsohave color.Instead, onehas tohave acollectionofgluons
whose colors add up to white. Such a collection forms an unstable
particlecalledaglueball.
The fact that confinement prevents one from observing an isolated
quark or gluon might seem to make the whole notion of quarks and
gluons as particles somewhat metaphysical. However, there is another
property of the strong nuclear force, called asymptotic freedom, that
makes the concept of quarks and gluons well defined. At normal
energies, the strong nuclear force is indeed strong, and it binds the
quarks tightly together. However, experiments with large particle
accelerators indicate that at high energies the strong force becomes
much weaker, and the quarks and gluons behave almost like free
particles. Fig. 5.2 shows a photograph of a collision between a high-
energy proton and antiproton. The success of the unification of the
electromagneticandweaknuclearforcesledtoanumberofattemptsto
combine these two forces with the strong nuclear force into what is
called a grand unified theory (or GUT). This title is rather an
exaggeration:theresultanttheoriesarenotallthatgrand,norarethey
fully unified, as they do not include gravity. Nor are they really
completetheories,becausetheycontainanumberofparameterswhose
valuescannotbepredictedfromthetheorybuthavetobechosentofit
inwithexperiment.Nevertheless,theymaybeasteptowardacomplete,
fully unified theory. The basic idea of GUTs is as follows: as was
mentionedabove,thestrongnuclearforcegetsweakerathighenergies.
Ontheotherhand,theelectromagneticandweakforces,whicharenot
asymptotically free, get stronger at high energies. At some very high
energy,calledthegrandunificationenergy,thesethreeforceswouldall
havethesamestrengthandsocouldjustbedifferentaspectsofasingle
force. The GUTs also predict that at this energy the different spin
matterparticles,likequarksandelectrons,wouldalsoallbeessentially
thesame,thusachievinganotherunification.
Thevalueofthegrandunificationenergyisnotverywellknown,but
it would probably have to be at least a thousand million million GeV.
The present generation of particle accelerators can collide particles at
energies of about one hundred GeV, and machines are planned that
would raise this to a few thousand GeV. But a machine that was
powerfulenoughtoaccelerateparticlestothegrandunificationenergy
wouldhavetobeasbigastheSolarSystem—andwouldbeunlikelyto
befundedinthepresenteconomicclimate.Thusitisimpossibletotest
grandunifiedtheoriesdirectlyinthelaboratory.However,justasinthe
case of the electromagnetic and weak unified theory, there are low-
energyconsequencesofthetheorythatcanbetested.
FIGURE5.2Aprotonandanantiprotoncollideathighenergy,producingacoupleof
almostfreequarks.
The most interesting of these is the prediction that protons, which
makeupmuchofthemassofordinarymatter,canspontaneouslydecay
intolighterparticlessuchasantielectrons.Thereasonthisispossibleis
that at the grand unification energy there is no essential difference
betweenaquarkandanantielectron.Thethreequarksinsideaproton
normally do not have enough energy to change into antielectrons, but
veryoccasionallyoneofthemmayacquiresufficientenergytomakethe
transitionbecausetheuncertaintyprinciplemeansthattheenergyofthe
quarksinsidetheprotoncannotbefixedexactly.Theprotonwouldthen
decay.Theprobabilityofaquarkgainingsufficientenergyissolowthat
one is likely to have to wait at least a million million million million
millionyears(1followedbythirtyzeros).Thisismuchlongerthanthe
timesincethebigbang,whichisameretenthousandmillionyearsorso
(1followedbytenzeros).Thusone mightthink that the possibilityof
spontaneousprotondecaycouldnotbetestedexperimentally.However,
onecanincreaseone’schancesofdetectingadecaybyobservingalarge
amount of matter containing a very large number of protons. (If, for
example, one observed a number of protons equal to 1 followed by
thirty-onezerosforaperiodofoneyear,onewouldexpect,accordingto
thesimplestGUT,toobservemorethanoneprotondecay.)
Anumberofsuchexperimentshavebeencarriedout,butnonehave
yielded definite evidence of proton or neutron decay. One experiment
usedeightthousandtonsofwaterandwasperformedintheMortonSalt
MineinOhio(toavoidothereventstakingplace,causedbycosmicrays,
thatmightbeconfusedwithprotondecay).Sincenospontaneousproton
decayhadbeenobservedduringtheexperiment,onecancalculatethat
theprobablelifeoftheprotonmustbegreaterthantenmillionmillion
million million million years (1 with thirty-one zeros). This is longer
than the lifetime predicted by the simplest grand unified theory, but
there are more elaborate theories in which the predicted lifetimes are
longer.Stillmoresensitiveexperimentsinvolvingevenlargerquantities
ofmatterwillbeneededtotestthem.
Eventhoughitisverydifficulttoobservespontaneousprotondecay,it
maybethatourveryexistenceisaconsequenceofthereverseprocess,
the production of protons, or more simply, of quarks, from an initial
situationinwhichtherewerenomorequarksthanantiquarks,whichis
themostnaturalwaytoimaginetheuniversestartingout.Matteronthe
earth is made up mainly of protons and neutrons, which in turn are
madeupofquarks.Therearenoantiprotonsorantineutrons,madeup
from antiquarks, except for a few that physicists produce in large
particleaccelerators.Wehaveevidencefromcosmicraysthatthesame
is true for all the matter in our galaxy: there are no antiprotons or
antineutrons apart from a small number that are produced as
particle/antiparticle pairs in high-energy collisions. If there were large
regions of antimatter in our galaxy, we would expect to observe large
quantitiesofradiationfromthe bordersbetweentheregionsof matter
and antimatter, where many particles would be colliding with their
antiparticles, annihilating each other and giving off high-energy
radiation.
Wehavenodirectevidenceastowhetherthematterinothergalaxies
ismadeupofprotonsandneutronsorantiprotonsandantineutrons,but
itmustbeoneortheother:therecannotbeamixtureinasinglegalaxy
because in that case we would again observe a lot of radiation from
annihilations. We therefore believe that all galaxies are composed of
quarks rather than antiquarks; it seems implausible that some galaxies
shouldbematterandsomeantimatter.
Whyshouldtherebesomanymorequarksthanantiquarks?Whyare
therenotequalnumbersofeach?Itiscertainlyfortunateforusthatthe
numbersareunequalbecause,iftheyhadbeenthesame,nearlyallthe
quarks and antiquarks would have annihilated each other in the early
universeandleftauniversefilledwithradiationbuthardlyanymatter.
There would then have been no galaxies, stars, or planets on which
human life could have developed. Luckily, grand unified theories may
provide an explanation of why the universe should now contain more
quarks than antiquarks, even if it started out with equal numbers of
each.Aswehaveseen,GUTsallowquarkstochangeintoantielectrons
athighenergy.Theyalsoallowthereverseprocesses,antiquarksturning
into electrons, and electrons and antielectrons turning into antiquarks
andquarks.Therewasatimeintheveryearlyuniversewhenitwasso
hot that the particle energies would have been high enough for these
transformationstotakeplace.Butwhyshouldthatleadtomorequarks
thanantiquarks?Thereasonisthatthelawsofphysicsarenotquitethe
sameforparticlesandantiparticles.
Up to 1956 it was believed that the laws of physics obeyed each of
three separate symmetries called C, P, and T. The symmetry C means
thatthelawsarethesameforparticlesandantiparticles.Thesymmetry
P means that the laws are the same for any situation and its mirror
image (the mirror image of a particle spinning in a right-handed
direction is one spinning in a left-handed direction). The symmetry T
means that if you reverse the direction of motion of all particles and
antiparticles,thesystemshouldgobacktowhatitwasatearliertimes;
in other words, the laws are the same in the forward and backward
directionsoftime.In1956twoAmericanphysicists,Tsung-DaoLeeand
ChenNingYang,suggestedthattheweakforcedoesnotinfactobeythe
symmetry P. In other words, the weak force would make the universe
developinadifferentwayfromthewayinwhichthemirrorimageof
theuniverse would develop.The same year,a colleague, Chien-Shiung
Wu,provedtheirpredictioncorrect.Shedidthisbyliningupthenuclei
ofradioactiveatomsinamagneticfield,sothattheywereallspinningin
thesamedirection,andshowedthattheelectronsweregivenoffmorein
onedirectionthananother.Thefollowingyear,LeeandYangreceived
theNobelPrizefortheiridea.Itwasalsofoundthattheweakforcedid
notobeythesymmetryC.Thatis,itwouldcauseauniversecomposedof
antiparticles to behave differently from our universe. Nevertheless, it
seemedthattheweakforcedidobeythecombinedsymmetryCP.That
is,theuniversewoulddevelopinthesamewayasitsmirrorimageif,in
addition,everyparticlewasswappedwithitsantiparticle!However,in
1964twomoreAmericans,J.W.CroninandValFitch,discoveredthat
eventheCPsymmetrywasnotobeyedinthedecayofcertainparticles
called K-mesons. Cronin and Fitch eventually received the Nobel Prize
fortheirworkin1980.(Alotofprizeshavebeenawardedforshowing
thattheuniverseisnotassimpleaswemighthavethought!)
Thereisamathematicaltheoremthatsaysthatanytheorythatobeys
quantum mechanics and relativity must always obey the combined
symmetryCPT.Inotherwords,theuniversewouldhavetobehavethe
same if one replaced particles by antiparticles, took the mirror image,
and also reversed the direction of time. But Cronin and Fitch showed
thatifonereplacesparticlesbyantiparticlesandtakesthemirrorimage,
but does not reverse the direction of time, then the universe does not
behave the same. The laws of physics, therefore, must change if one
reversesthedirectionoftime—theydonotobeythesymmetryT.
Certainly the early universe does not obey the symmetry T: as time
runs forward the universe expands—if it ran backward, the universe
would be contracting. And since there are forces that do not obey the
symmetryT,itfollowsthatastheuniverseexpands,theseforcescould
cause more antielectrons to turn into quarks than electrons into
antiquarks.Then, astheuniverse expandedand cooled, theantiquarks
wouldannihilatewiththequarks,butsincetherewouldbemorequarks
thanantiquarks,asmallexcessofquarkswouldremain.Itisthesethat
make up the matter we see today and out of which we ourselves are
made. Thus our very existence could be regarded as a confirmation of
grandunifiedtheories,thoughaqualitativeoneonly;theuncertainties
aresuchthatonecannotpredictthenumbersofquarksthatwillbeleft
aftertheannihilation,orevenwhetheritwouldbequarksorantiquarks
thatwouldremain. (Haditbeen anexcessof antiquarks,however,we
wouldsimplyhavenamedantiquarksquarks,andquarksantiquarks.)
Grandunifiedtheoriesdonotincludetheforceofgravity.Thisdoes
notmattertoomuch,becausegravityissuchaweakforcethatitseffects
canusuallybeneglectedwhenwearedealingwithelementaryparticles
or atoms. However, the fact that it is both long range and always
attractive means that its effects all add up. So for a sufficiently large
number of matter particles, gravitational forces can dominate over all
otherforces.Thisiswhyitisgravitythatdeterminestheevolutionofthe
universe.Evenforobjectsthesizeofstars,theattractiveforceofgravity
canwinoveralltheotherforcesandcausethestartocollapse.Mywork
inthe1970sfocusedontheblackholesthatcanresultfromsuchstellar
collapseandtheintensegravitationalfieldsaroundthem.Itwasthisthat
led to the first hints of how the theories of quantum mechanics and
generalrelativitymightaffecteachother—a glimpseofthe shapeofa
quantumtheoryofgravityyettocome.
T
CHAPTER6
BLACKHOLES
hetermblackholeisofveryrecentorigin.Itwascoinedin1969by
theAmericanscientistJohnWheelerasagraphic descriptionof an
idea that goes back at least two hundred years, to a time when there
weretwotheoriesaboutlight:one,whichNewtonfavored,wasthatit
wascomposedofparticles;theotherwasthatitwasmadeofwaves.We
now know that really both theories are correct. By the wave/particle
dualityofquantummechanics,lightcanberegardedasbothawaveand
aparticle.Underthetheorythatlightismadeupofwaves,itwasnot
clear how it would respond to gravity. But if light is composed of
particles,onemightexpectthemtobeaffectedbygravityinthesame
waythat cannonballs, rockets,andplanets are. Atfirstpeople thought
thatparticlesoflighttraveledinfinitelyfast,sogravitywouldnothave
been able to slow them down, but the discovery by Roemer that light
travels at a finite speed meant that gravity might have an important
effect.
Onthisassumption,aCambridgedon,JohnMichell,wroteapaperin
1783 in the Philosophical Transactions of the Royal Society of London in
which he pointed out that a star that was sufficiently massive and
compactwouldhavesuchastronggravitationalfieldthatlightcouldnot
escape:anylightemittedfromthesurfaceofthestarwouldbedragged
back by the star’s gravitational attraction before it could get very far.
Michellsuggestedthattheremightbealargenumberofstarslikethis.
Althoughwewouldnotbeabletoseethembecausethelightfromthem
would not reach us, we would still feel their gravitational attraction.
Suchobjectsarewhatwenowcallblackholes,becausethatiswhatthey
are: black voids in space. A similar suggestion was made a few years
later by the French scientist the Marquis de Laplace, apparently
independently of Michell. Interestingly enough, Laplace included it in
onlythefirst andsecondeditions ofhis bookThe System of the World,
andleftitoutoflatereditions;perhapshedecidedthatitwasacrazy
idea. (Also, the particle theory of light went out of favor during the
nineteenthcentury;itseemedthateverythingcouldbeexplainedbythe
wave theory, and according to the wave theory, it was not clear that
lightwouldbeaffectedbygravityatall.)
In fact, it is not really consistent to treat light like cannonballs in
Newton’s theory of gravity because the speed of light is fixed. (A
cannonballfiredupwardfromtheearthwillbesloweddownbygravity
andwilleventuallystopandfallback;aphoton,however,mustcontinue
upward at a constant speed. How then can Newtonian gravity affect
light?) A consistent theory of how gravity affects light did not come
alonguntilEinsteinproposedgeneralrelativityin1915.Andeventhen
itwasalongtimebeforetheimplicationsofthetheoryformassivestars
wereunderstood.
To understand how a black hole might be formed, we first need an
understandingofthelifecycleofastar.Astarisformedwhenalarge
amountofgas(mostlyhydrogen)startstocollapseinonitselfduetoits
gravitationalattraction.Asitcontracts,theatomsofthegascollidewith
eachothermoreandmorefrequentlyandatgreaterandgreaterspeeds
—the gas heats up. Eventually, the gas will be so hot that when the
hydrogen atoms collide they no longer bounce off each other, but
instead coalesce to form helium. The heat released in this reaction,
whichislikeacontrolledhydrogenbombexplosion,iswhatmakesthe
star shine. This additional heat also increases the pressure of the gas
untilitissufficienttobalancethegravitationalattraction,andthe gas
stopscontracting.Itisabitlikeaballoon—thereisabalancebetween
the pressure of the air inside, which is trying to make the balloon
expand, and the tension in the rubber, which is trying to make the
balloonsmaller.Starswillremainstablelikethisforalongtime,with
heat from the nuclear reactions balancing the gravitational attraction.
Eventually, however, the star will run out of its hydrogen and other
nuclear fuels. Paradoxically, the more fuel a star starts off with, the
sooner it runs out. This is because the more massive the star is, the
hotter it needs to be to balance its gravitational attraction. And the
hotterit is,the fasterit willuse upits fuel.Oursun hasprobably got
enough fuel for another five thousand million years or so, but more
massive stars can use up their fuel in as little as one hundred million
years,muchlessthantheageoftheuniverse.Whenastarrunsoutof
fuel,itstartstocooloffandsotocontract.Whatmighthappentoitthen
wasfirstunderstoodonlyattheendofthe1920s.
In 1928 an Indian graduate student, Subrahmanyan Chandrasekhar,
setsailforEnglandtostudyatCambridgewiththeBritishastronomer
SirArthurEddington,anexpertongeneralrelativity.(Accordingtosome
accounts, a journalist told Eddington in the early 1920s that he had
heardtherewereonlythreepeopleintheworldwhounderstoodgeneral
relativity.Eddingtonpaused,thenreplied,“Iamtryingtothinkwhothe
thirdpersonis.”)DuringhisvoyagefromIndia,Chandrasekharworked
outhowbigastarcouldbeandstillsupportitselfagainstitsowngravity
afterithadusedupallitsfuel.Theideawasthis:whenthestarbecomes
small,thematterparticlesgetveryneareachother,andsoaccordingto
the Pauli exclusion principle, they must have very different velocities.
Thismakesthemmoveawayfromeachotherandsotendstomakethe
starexpand.Astarcanthereforemaintainitselfataconstantradiusbya
balancebetween theattraction of gravity and the repulsion that arises
from the exclusion principle, just as earlier in its life gravity was
balancedbytheheat.
Chandrasekharrealized,however,thatthereisalimittotherepulsion
thattheexclusionprinciplecanprovide.Thetheoryofrelativitylimits
themaximumdifferenceinthevelocitiesofthematterparticlesinthe
startothespeedoflight.Thismeansthatwhenthestargotsufficiently
dense,therepulsioncausedbytheexclusionprinciplewouldbelessthan
the attraction of gravity. Chandrasekhar calculated that a cold star of
morethanaboutoneandahalftimesthemassofthesunwouldnotbe
abletosupportitselfagainstitsowngravity.(Thismassisnowknownas
theChandrasekharlimit.)Asimilardiscoverywasmadeaboutthesame
timebytheRussianscientistLevDavidovichLandau.
Thishadseriousimplicationsfortheultimatefateofmassivestars.Ifa
star’smass is less than the Chandrasekharlimit,it can eventually stop
contractingandsettledowntoapossiblefinalstateasa“whitedwarf”
witharadiusofafewthousandmilesandadensityofhundredsoftons
per cubic inch. A white dwarf is supported by the exclusion principle
repulsionbetweentheelectronsinitsmatter.Weobservealargenumber
ofthesewhitedwarfstars.Oneofthefirsttobediscoveredisastarthat
isorbitingaroundSirius,thebrighteststarinthenightsky.
Landau pointed out that there was another possible final state for a
star,alsowithalimitingmassofaboutoneortwotimesthemassofthe
sun but much smaller even than a white dwarf. These stars would be
supported by the exclusion principle repulsion between neutrons and
protons, rather than between electrons. They were therefore called
neutronstars.Theywouldhave aradiusofonly tenmilesorso anda
densityofhundredsofmillionsoftonspercubicinch.Atthetimethey
were first predicted, there was no way that neutron stars could be
observed.Theywerenotactuallydetecteduntilmuchlater.
StarswithmassesabovetheChandrasekharlimit,ontheotherhand,
have a big problem when they come to the end of their fuel. In some
casestheymayexplodeormanagetothrowoffenoughmattertoreduce
their mass below the limit and so avoid catastrophic gravitational
collapse, but it was difficult to believe that this always happened, no
matterhowbigthestar.Howwoulditknowthatithadtoloseweight?
Andevenifeverystarmanagedtoloseenoughmasstoavoidcollapse,
whatwouldhappenifyouaddedmoremasstoawhitedwarforneutron
star to take it over the limit? Would it collapse to infinite density?
Eddington was shocked by that implication, and he refused to believe
Chandrasekhar’s result. Eddington thought it was simply not possible
thatastarcouldcollapsetoapoint.Thiswastheviewofmostscientists:
Einsteinhimselfwroteapaperinwhichheclaimedthatstarswouldnot
shrink to zero size. The hostility of other scientists, particularly
Eddington,hisformerteacherandtheleadingauthorityonthestructure
ofstars,persuadedChandrasekhartoabandonthislineofworkandturn
instead to other problems in astronomy, such as the motion of star
clusters. However, when he was awarded the Nobel Prize in 1983, it
was,atleastinpart,forhisearlyworkonthelimitingmassofcoldstars.
Chandrasekharhadshownthattheexclusionprinciplecouldnothalt
the collapse of a star more massive than the Chandrasekhar limit, but
the problem of understanding what would happen to such a star,
according to general relativity, was first solved by a young American,
RobertOppenheimer,in1939.Hisresult,however,suggestedthatthere
wouldbenoobservationalconsequencesthatcouldbedetectedbythe
telescopesoftheday.ThenWorldWarIIintervenedandOppenheimer
himselfbecamecloselyinvolvedintheatombombproject.Afterthewar
the problem of gravitational collapse was largely forgotten as most
scientistsbecamecaught upin whathappens onthe scaleof theatom
and its nucleus. In the 1960s, however, interest in the large-scale
problemsofastronomyandcosmologywasrevivedbyagreatincreasein
the number and range of astronomical observations brought about by
the application of modern technology. Oppenheimer’s work was then
rediscoveredandextendedbyanumberofpeople.
ThepicturethatwenowhavefromOppenheimer’sworkisasfollows.
The gravitational field of the star changes the paths of light rays in
space-time from what they would have been had the star not been
present.Thelightcones,whichindicatethepathsfollowedinspaceand
timebyflashesoflightemittedfromtheirtips,arebentslightlyinward
nearthesurfaceofthestar.Thiscanbeseeninthebendingoflightfrom
distantstarsobservedduringaneclipseofthesun.Asthestarcontracts,
thegravitationalfieldatitssurfacegetsstrongerandthelightconesget
bentinwardmore.Thismakesitmoredifficultforlightfromthestarto
escape, and the light appears dimmer and redder to an observer at a
distance. Eventually, when the star has shrunk to a certain critical
radius,thegravitationalfieldatthesurfacebecomessostrongthatthe
lightconesarebentinwardsomuchthatlightcannolongerescape(Fig.
6.1).Accordingtothetheoryofrelativity,nothingcantravelfasterthan
light.Thusiflightcannotescape,neithercananythingelse;everything
isdraggedbackbythegravitationalfield.Soonehasasetofevents,a
regionofspace-time,fromwhichitisnotpossibletoescapetoreacha
distant observer. This region is what we now call a black hole. Its
boundaryiscalledtheeventhorizonanditcoincideswiththepathsof
lightraysthatjustfailtoescapefromtheblackhole.
In order to understand what you would see if you were watching a
star collapse to form a black hole, one has to remember that in the
theoryofrelativitythereisnoabsolutetime.Eachobserverhashisown
measureoftime.Thetimeforsomeoneonastarwillbedifferentfrom
thatforsomeoneatadistance,becauseofthegravitationalfieldofthe
star.Supposeanintrepidastronautonthesurfaceofthecollapsingstar,
collapsing inward with it, sent a signal every second, according to his
watch, to his spaceship orbiting about the star. At some time on his
watch, say 11:00, the star would shrink below the critical radius at
whichthegravitationalfieldbecomessostrongnothingcanescape,and
hissignalswouldnolongerreachthespaceship.As11:00approached,
his companions watching from the spaceship would find the intervals
betweensuccessivesignalsfromtheastronautgettinglongerandlonger,
butthiseffectwouldbeverysmallbefore10:59:59.Theywouldhaveto
wait only very slightly more than a second between the astronaut’s
10:59:58signalandtheonethathesentwhenhiswatchread10:59:59,
but they would have to wait forever for the 11:00 signal. The light
wavesemittedfromthesurfaceofthestarbetween10:59:59and11:00,
bytheastronaut’swatch,wouldbespreadoutoveraninfiniteperiodof
time,asseenfromthespaceship.Thetimeintervalbetweenthearrival
ofsuccessivewavesatthespaceshipwouldgetlongerandlonger,sothe
light from the star would appear redder and redder and fainter and
fainter.Eventually,thestarwouldbesodimthatitcouldnolongerbe
seenfromthespaceship:allthatwouldbeleftwouldbeablackholein
space.Thestarwould,however,continuetoexertthesamegravitational
force on the spaceship, which would continue to orbit the black hole.
Thisscenarioisnotentirelyrealistic,however,becauseofthefollowing
problem.Gravitygetsweakerthe fartheryouarefrom thestar,sothe
gravitational force on our intrepid astronaut’s feet would always be
greaterthantheforceonhishead.Thisdifferenceintheforceswould
stretchourastronautoutlikespaghettiortearhimapartbeforethestar
hadcontractedtothecriticalradiusatwhichtheeventhorizonformed!
However,webelievethattherearemuchlargerobjectsintheuniverse,
likethecentralregionsofgalaxies,thatcanalsoundergogravitational
collapsetoproduceblackholes;anastronautononeofthesewouldnot
betornapartbeforetheblackholeformed.Hewouldnot,infact,feel
anything special as he reached the critical radius, and could pass the
pointofnoreturnwithoutnoticingit.However,withinjustafewhours,
as the region continued to collapse, the difference in the gravitational
forces on his head and his feet would become so strong that again it
wouldtearhimapart.
FIGURE6.1
The work that Roger Penrose and I did between 1965 and 1970
showedthat,accordingtogeneralrelativity,theremustbeasingularity
ofinfinitedensityandspace-timecurvaturewithinablackhole.Thisis
ratherlike the big bang atthebeginning of time, only it wouldbe an
endoftimeforthecollapsingbodyandtheastronaut.Atthissingularity
the laws of science and our ability to predict the future would break
down. However, any observer who remained outside the black hole
would not be affected by this failure of predictability, because neither
light nor any other signal could reach him from the singularity. This
remarkable fact led Roger Penrose to propose the cosmic censorship
hypothesis, which might be paraphrased as “God abhors a naked
singularity.”Inotherwords,thesingularitiesproducedbygravitational
collapseoccuronlyinplaces,likeblackholes,wheretheyaredecently
hiddenfromoutside viewby anevent horizon.Strictly, thisis whatis
knownasthe weakcosmiccensorshiphypothesis:it protectsobservers
who remain outside the black hole from the consequences of the
breakdown of predictability that occurs at the singularity, but it does
nothingatallforthepoorunfortunateastronautwhofallsintothehole.
There are some solutions of the equations of general relativity in
whichitispossibleforourastronauttoseeanakedsingularity:hemay
be able to avoid hitting the singularity and instead fall through a
“wormhole”andcomeoutinanotherregionoftheuniverse.Thiswould
offergreatpossibilitiesfortravelinspaceandtime,butunfortunatelyit
seems that these solutions may all be highly unstable; the least
disturbance,suchasthepresenceofanastronaut,maychangethemso
that the astronaut could not see the singularity until he hit it and his
timecametoanend.Inotherwords,thesingularitywouldalwaysliein
his future and never in his past. The strong version of the cosmic
censorshiphypothesisstatesthatinarealisticsolution,thesingularities
would always lie either entirely in the future (like the singularities of
gravitational collapse) or entirely in the past (like the big bang). I
strongly believe in cosmic censorship so I bet Kip Thorne and John
Preskill of Cal Tech that it would always hold. I lost the bet on a
technicality because examples were produced of solutions with a
singularitythatwasvisiblefromalongwayaway.SoIhadtopayup,
which according to the terms of the bet meant I had to clothe their
nakedness.ButIcanclaimamoralvictory.Thenakedsingularitieswere
unstable:theleastdisturbancewouldcausethemeithertodisappearor
to be hidden behind an event horizon. So they would not occur in
realisticsituations.
The event horizon, the boundary of the region of space-time from
whichitisnotpossibletoescape,actsratherlikeaone-waymembrane
around the black hole: objects, such as unwary astronauts, can fall
throughtheeventhorizonintotheblackhole,butnothingcaneverget
out of the black hole through the event horizon. (Remember that the
eventhorizonisthepathinspace-timeoflightthatistryingtoescape
fromtheblackhole,andnothingcantravelfasterthanlight.)Onecould
wellsayoftheeventhorizonwhatthepoetDantesaidoftheentranceto
Hell:“Allhopeabandon,yewhoenterhere.”Anythingoranyonewho
falls through the event horizon will soon reach the region of infinite
densityandtheendoftime.
General relativity predicts that heavy objects that are moving will
cause the emission of gravitational waves, ripples in the curvature of
spacethattravelatthespeedoflight.Thesearesimilartolightwaves,
whichareripplesoftheelectromagneticfield,buttheyaremuchharder
todetect.Theycanbeobservedbytheveryslightchangeinseparation
theyproducebetweenneighboringfreelymovingobjects.Anumberof
detectors are being built in the United States, Europe, and Japan that
will measure displacements of one part in a thousand million million
million(1withtwenty-onezerosafterit),orlessthanthenucleusofan
atomoveradistanceoftenmiles.
Likelight,gravitationalwavescarryenergyawayfromtheobjectsthat
emitthem. One would therefore expect a systemof massive objects to
settledowneventuallytoastationarystate,becausetheenergyinany
movement would be carried away by the emission of gravitational
waves.(It is rather like droppinga cork into water: at firstit bobs up
and down a great deal, but as the ripples carry away its energy, it
eventually settles down to a stationary state.) For example, the
movementoftheearthinitsorbitroundthesunproducesgravitational
waves.The effectof the energyloss will beto changetheorbit of the
earthsothatgraduallyitgetsnearerandnearertothesun,eventually
collideswithit,andsettlesdowntoastationarystate.Therateofenergy
lossinthecaseoftheearthandthesunisverylow—aboutenoughto
run a small electric heater. This means it will take about a thousand
millionmillionmillionmillionyearsfortheearthtorunintothesun,so
there’s no immediate cause for worry! The change in the orbit of the
earthistooslowtobeobserved,butthissameeffecthasbeenobserved
overthepastfewyearsoccurringinthesystemcalledPSR1913+16
(PSRstandsfor“pulsar,”aspecialtypeofneutronstarthatemitsregular
pulsesofradiowaves).Thissystemcontainstwoneutronstarsorbiting
each other, and the energy they are losing by the emission of
gravitationalwavesiscausingthemtospiralintowardeachother.This
confirmationofgeneralrelativitywonJ.H.TaylorandR.A.Hulsethe
NobelPrizein1993.Itwilltakeaboutthreehundredmillionyearsfor
them to collide. Just before they do, they will be orbiting so fast that
theywillemitenoughgravitationalwavesfordetectorslikeLIGOtopick
up.
During the gravitational collapse of a star to form a black hole, the
movementswouldbemuchmorerapid,sotherateatwhichenergyis
carriedawaywouldbemuchhigher.Itwouldthereforenotbetoolong
beforeitsettleddowntoastationarystate.Whatwouldthisfinalstage
looklike?Onemightsupposethatitwoulddependonallthecomplex
featuresofthestarfromwhichithadformed—notonlyitsmassandrate
ofrotation,butalsothedifferentdensitiesofvarious partsofthestar,
and the complicated movements of the gases within the star. And if
blackholeswereasvariedastheobjectsthatcollapsedtoformthem,it
might be very difficult to make any predictions about black holes in
general.
In 1967, however, the study of black holes was revolutionized by
WernerIsrael,aCanadianscientist(whowasborninBerlin,broughtup
inSouthAfrica,andtookhisdoctoraldegreeinIreland).Israelshowed
that, according to general relativity, non-rotating black holes must be
verysimple;theywereperfectlyspherical,theirsizedependedonlyon
their mass, and any two such black holes with the same mass were
identical. They could, in fact, be described by a particular solution of
Einstein’s equations that had been known since 1917, found by Karl
Schwarzschild shortly after the discovery of general relativity. At first
manypeople,includingIsraelhimself,arguedthatsinceblackholeshad
tobeperfectlyspherical,ablackholecouldonlyformfromthecollapse
of a perfectly spherical object. Any real star—which would never be
perfectly spherical—could therefore only collapse to form a naked
singularity.
Therewas,however,adifferentinterpretationofIsrael’sresult,which
wasadvocatedbyRogerPenroseandJohnWheelerinparticular.They
argued that the rapid movements involved in a star’s collapse would
meanthatthegravitationalwavesitgaveoffwouldmakeitevermore
spherical, and by the time it had settled down to a stationary state, it
would be precisely spherical. According to this view, any non-rotating
star,howevercomplicateditsshapeandinternalstructure,wouldendup
aftergravitationalcollapseasaperfectlysphericalblackhole,whosesize
woulddependonlyonitsmass.Furthercalculationssupportedthisview,
anditsooncametobeadoptedgenerally.
Israel’s result dealt with the case of black holes formed from non-
rotatingbodiesonly.In1963,RoyKerr,aNewZealander,foundasetof
solutions of the equations of general relativity that described rotating
blackholes.These“Kerr”blackholesrotateataconstantrate,theirsize
and shape depending only on their mass and rate of rotation. If the
rotation is zero, the black hole is perfectly round and the solution is
identical to the Schwarzschild solution. If the rotation is non-zero, the
blackholebulgesoutwardnearitsequator(justastheearthorthesun
bulgeduetotheirrotation),andthefasteritrotates,themoreitbulges.
So,toextendIsrael’sresulttoincluderotatingbodies,itwasconjectured
that any rotating body that collapsed to form a black hole would
eventually settle down to a stationary state described by the Kerr
solution.
In1970acolleagueandfellowresearchstudentofmineatCambridge,
Brandon Carter, took the first step toward proving this conjecture. He
showed that, provided a stationary rotating black hole had an axis of
symmetry,likeaspinningtop,itssizeandshapewoulddependonlyon
itsmassandrateofrotation.Then,in1971,Iprovedthatanystationary
rotatingblackholewouldindeedhavesuchanaxisofsymmetry.Finally,
in1973,DavidRobinsonatKingsCollege,London,usedCarter’sandmy
resultstoshowthattheconjecturehadbeencorrect:suchablackhole
hadindeedtobetheKerrsolution.Soaftergravitationalcollapseablack
holemustsettledownintoastateinwhichitcouldberotating,butnot
pulsating.Moreover,itssizeandshapewoulddependonlyonitsmass
and rate of rotation, and not on the nature of the body that had
collapsedtoformit.Thisresultbecameknownbythemaxim:“Ablack
hole has no hair.” The “no hair” theorem is of great practical
importance, because it so greatly restricts the possible types of black
holes. One can therefore make detailed models of objects that might
contain black holes and compare the predictions of the models with
observations. It also means that a very large amount of information
about the body that has collapsed must be lost when a black hole is
formed,becauseafterwardallwecanpossiblymeasureaboutthebodyis
itsmassandrateofrotation.Thesignificanceofthiswillbeseeninthe
nextchapter.
Black holes are one of only a fairly small number of cases in the
historyofscienceinwhichatheorywasdevelopedingreatdetailasa
mathematical model before there was any evidence from observations
that it was correct. Indeed, this used to be the main argument of
opponentsofblackholes:howcouldonebelieveinobjectsforwhichthe
onlyevidencewascalculationsbasedonthedubioustheoryofgeneral
relativity? In 1963, however, Maarten Schmidt, an astronomer at the
Palomar Observatory in California, measured the red shift of a faint
starlikeobjectinthedirectionofthesourceofradiowavescalled3C273
(thatis, source number 273 in the thirdCambridgecatalogue of radio
sources).Hefounditwastoolargetobecausedbyagravitationalfield:
ifithadbeenagravitationalredshift,theobjectwouldhavetobeso
massiveandsoneartousthatitwoulddisturbtheorbitsofplanetsin
theSolarSystem.Thissuggestedthattheredshiftwasinsteadcausedby
theexpansionoftheuniverse,which,inturn,meantthattheobjectwas
averylongdistanceaway.Andtobevisibleatsuchagreatdistance,the
object must be very bright, must, in other words, be emitting a huge
amountofenergy.Theonlymechanismthatpeoplecouldthinkofthat
would produce such large quantities of energy seemed to be the
gravitationalcollapsenotjustofastarbutofawholecentralregionofa
galaxy. A number of other similar “quasi-stellar objects,” or quasars,
havebeendiscovered,allwithlargeredshifts.Buttheyarealltoofar
away and therefore too difficult to observe to provide conclusive
evidenceofblackholes.
Furtherencouragementfortheexistenceofblackholescamein1967
with the discovery by a research student at Cambridge, Jocelyn Bell-
Burnell,ofobjectsintheskythatwereemittingregularpulsesofradio
waves. At first Bell and her supervisor, Antony Hewish, thought they
mighthavemadecontactwithanaliencivilizationinthegalaxy!Indeed,
attheseminaratwhichtheyannouncedtheirdiscovery,Irememberthat
theycalledthefirstfoursourcestobefoundLGM1–4,LGMstandingfor
“LittleGreenMen.”Intheend,however,theyandeveryoneelsecameto
the less romantic conclusion that these objects, which were given the
name pulsars, were in fact rotating neutron stars that were emitting
pulsesofradiowavesbecauseofacomplicatedinteractionbetweentheir
magneticfieldsandsurroundingmatter.Thiswasbadnewsforwriters
of space westerns, but very hopeful for the small number of us who
believed in black holes at that time: it was the first positive evidence
thatneutronstarsexisted.Aneutronstarhasaradiusofabouttenmiles,
onlyafewtimesthecriticalradiusatwhichastarbecomesablackhole.
If a star could collapse to such a small size, it is not unreasonable to
expectthatother starscouldcollapse toevensmaller sizeandbecome
blackholes.
Howcouldwehopetodetectablackhole,asbyitsverydefinitionit
doesnotemitanylight?Itmightseemabitlikelookingforablackcat
inacoalcellar.Fortunately,thereisaway.AsJohnMichellpointedout
inhispioneeringpaperin1783,ablackholestillexertsagravitational
force on nearby objects. Astronomers have observed many systems in
whichtwostarsorbitaroundeachother,attractedtowardeachotherby
gravity.Theyalsoobservesystemsinwhichthereisonlyonevisiblestar
thatisorbitingaroundsomeunseencompanion.Onecannot,ofcourse,
immediately conclude that the companion is a black hole: it might
merely be a star that is too faint to be seen. However, some of these
systems,liketheonecalledCygnusX-l(Fig.6.2),arealsostrongsources
ofXrays.Thebestexplanationforthisphenomenonisthatmatterhas
been blown off the surface of the visible star. As it falls toward the
unseen companion, it develops a spiral motion (rather like water
runningoutofabath),anditgetsveryhot,emittingXrays(Fig.6.3).
Forthismechanismtowork,theunseenobjecthastobeverysmall,like
awhitedwarf,neutronstar, orblack hole.From theobserved orbitof
the visible star, one can determine the lowest possible mass of the
unseenobject.InthecaseofCygnusX-l,thisisaboutsixtimesthemass
ofthesun,which,accordingtoChandrasekhar’sresult,istoogreatfor
theunseenobjecttobeawhitedwarf.Itisalsotoolargeamasstobea
neutronstar.Itseems,therefore,thatitmustbeablackhole.
There are other models to explain Cygnus X-l that do not include a
blackhole,buttheyareallratherfar-fetched.Ablackholeseemstobe
the only really natural explanation of the observations. Despite this, I
hadabetwithKipThorneoftheCaliforniaInstituteofTechnologythat
in fact Cygnus X-l does not contain a black hole! This was a form of
insurancepolicyforme.Ihavedonealotofworkonblackholes,andit
wouldallbewastedifitturnedoutthatblackholesdonotexist.Butin
thatcase,Iwouldhavetheconsolationofwinningmybet,whichwould
bring me four years of the magazine Private Eye. In fact, although the
situationwithCygnusX-lhasnotchangedmuchsincewemadethebet
in1975,thereisnowsomuchotherobservationalevidenceinfavorof
black holes that I have conceded the bet. I paid the specified penalty,
whichwasaone-yearsubscriptiontoPenthouse,totheoutrageofKip’s
liberatedwife.
FIGURE6.2ThebrighterofthetwostarsnearthecenterofthephotographisCygnusX-l,
whichisthoughttoconsistofablackholeandanormalstar,orbitingaroundeachother.
We also now have evidence for several other black holes in systems
likeCygnusX-linourgalaxyandintwoneighboringgalaxiescalledthe
Magellanic Clouds. The number of black holes, however, is almost
certainly very much higher; in the long history of the universe, many
starsmusthaveburnedalltheirnuclearfuelandhavehadtocollapse.
Thenumberofblackholesmaywellbegreatereventhanthenumberof
visible stars, which totals about a hundred thousand million in our
galaxyalone.Theextragravitationalattractionofsuchalargenumber
ofblackholescouldexplainwhyourgalaxyrotatesattherateitdoes:
themassofthevisiblestarsis insufficienttoaccountforthis.Wealso
havesomeevidencethatthereisamuchlargerblackhole,withamass
ofaboutahundredthousandtimesthatofthesun,atthecenterofour
galaxy. Stars in the galaxy that come too near this black hole will be
tornapartbythedifferenceinthegravitationalforcesontheirnearand
farsides.Theirremains,andgasthatisthrownoffotherstars,willfall
towardtheblackhole.AsinthecaseofCygnusX-l,thegaswillspiral
inwardandwillheatup,thoughnotasmuchasinthatcase.Itwillnot
gethotenoughtoemitXrays,butitcouldaccountfortheverycompact
sourceofradiowavesandinfraredraysthatisobservedatthegalactic
center.
FIGURE6.3
Itisthoughtthatsimilarbutevenlargerblackholes,withmassesof
aboutahundredmilliontimesthemassofthesun,occuratthecenters
ofquasars.Forexample,observationswiththeHubbletelescopeofthe
galaxyknownasM87revealthatitcontainsadiskofgas130light-years
across rotating about a central object two thousand million times the
massofthesun.Thiscanonlybeablackhole.Matterfallingintosucha
supermassiveblackholewouldprovidetheonlysourceofpowergreat
enough to explain the enormous amounts of energy that these objects
areemitting.Asthematterspiralsintotheblackhole,itwouldmakethe
blackholerotateinthesamedirection,causingittodevelopamagnetic
fieldrather likethat of theearth. Very high-energyparticles wouldbe
generated near the black hole by the in-falling matter. The magnetic
field would be so strong that it could focus these particles into jets
ejectedoutwardalongtheaxisofrotationoftheblackhole,thatis,in
thedirectionsofitsnorthandsouthpoles.Suchjetsareindeedobserved
in a number of galaxies and quasars. One can also consider the
possibilitythattheremightbeblackholeswithmassesmuchlessthan
thatofthesun. Suchblackholescouldnotbeformedbygravitational
collapse,becausetheirmassesarebelowtheChandrasekharmasslimit:
starsofthislowmasscansupportthemselvesagainsttheforceofgravity
evenwhentheyhaveexhaustedtheirnuclearfuel.Low-massblackholes
couldformonlyifmatterwascompressedtoenormousdensitiesbyvery
large external pressures. Such conditions could occur in a very big
hydrogenbomb:thephysicistJohnWheeleroncecalculatedthatifone
tookalltheheavywaterinalltheoceansoftheworld,onecouldbuilda
hydrogenbombthatwouldcompressmatteratthecentersomuchthata
blackholewouldbecreated.(Ofcourse,therewouldbenooneleftto
observe it!) A more practical possibility is that such low-mass black
holesmighthavebeenformedinthehightemperaturesandpressuresof
theveryearlyuniverse.Blackholeswouldhavebeenformedonlyifthe
earlyuniversehadnotbeenperfectlysmoothanduniform,becauseonly
asmallregionthatwasdenserthanaveragecouldbecompressedinthis
waytoformablackhole.Butweknowthattheremusthavebeensome
irregularities,becauseotherwisethematterintheuniversewouldstillbe
perfectly uniformly distributed at the present epoch, instead of being
clumpedtogetherinstarsandgalaxies.
Whether the irregularities required to account for stars and galaxies
wouldhaveledtotheformationofasignificantnumberof“primordial”
blackholesclearlydependsonthedetailsoftheconditionsintheearly
universe. So if we could determine how many primordial black holes
therearenow,wewouldlearnalotabouttheveryearlystagesofthe
universe. Primordial black holes with masses more than a thousand
milliontons(the massof alargemountain)could bedetected onlyby
theirgravitationalinfluenceonother,visiblematterorontheexpansion
of the universe. However, as we shall learn in the next chapter, black
holesarenotreallyblackafterall:theyglowlikeahot body,andthe
smaller they are, the more they glow. So, paradoxically, smaller black
holesmightactuallyturnouttobeeasiertodetectthanlargeones!
B
CHAPTER7
BLACKHOLES
AIN’TSOBLACK
efore 1970, my research on general relativity had concentrated
mainlyonthequestionofwhetherornottherehadbeenabigbang
singularity.However,oneeveninginNovemberthatyear,shortlyafter
thebirthofmydaughter,Lucy,IstartedtothinkaboutblackholesasI
wasgettingintobed.Mydisabilitymakesthisratheraslowprocess,soI
hadplentyoftime.Atthatdatetherewasnoprecisedefinitionofwhich
pointsinspace-timelayinsideablackholeandwhichlayoutside.Ihad
alreadydiscussedwithRogerPenrosetheideaofdefiningablackholeas
the set of events from which it was not possible to escape to a large
distance,whichisnowthegenerallyaccepteddefinition.Itmeansthat
theboundaryoftheblackhole,theeventhorizon,isformedbythelight
raysthatjustfailtoescapefromtheblackhole,hoveringforeverjuston
theedge(Fig.7.1).Itisabitlikerunningawayfromthepoliceandjust
managingtokeeponestepaheadbutnotbeingabletogetclearaway!
Suddenly I realized that the paths of these light rays could never
approach one another. If they did, they must eventually run into one
another.Itwouldbelikemeetingsomeoneelserunningawayfromthe
policeintheoppositedirection—youwouldbothbecaught!(Or,inthis
case,fallintoablackhole.)Butiftheselightrayswereswallowedupby
theblack hole,then theycould nothavebeen onthe boundaryof the
blackhole.Sothepathsoflightraysintheeventhorizonhadalwaysto
bemovingparallelto,orawayfrom,eachother.Anotherwayofseeing
thisisthattheeventhorizon,theboundaryoftheblackhole,islikethe
edgeofa shadow—theshadow ofimpending doom.If youlook atthe
shadowcastbyasourceatagreatdistance,suchasthesun,youwillsee
thattheraysoflightintheedgearenotapproachingeachother.
FIGURE7.1
Iftheraysoflightthatformtheeventhorizon,theboundaryofthe
blackhole,canneverapproacheachother,theareaoftheeventhorizon
mightstaythesameorincreasewithtime,butitcouldneverdecrease
becausethatwouldmeanthatat leastsomeof theraysoflightinthe
boundary would have to be approaching each other. In fact, the area
would increase whenever matter or radiation fell into the black hole
(Fig.7.2).Oriftwoblackholescollidedandmergedtogethertoforma
singleblackhole,theareaof theeventhorizon ofthefinalblackhole
would be greater than or equal to the sum of the areas of the event
horizons of the original black holes (Fig. 7.3). This nondecreasing
propertyoftheeventhorizon’sareaplacedanimportantrestrictionon
thepossiblebehaviorofblackholes.Iwassoexcitedwithmydiscovery
thatIdidnotgetmuchsleepthatnight.ThenextdayIrangupRoger
Penrose.Heagreedwithme.Ithink,infact,thathehadbeenawareof
thispropertyofthearea.However,hehadbeenusingaslightlydifferent
definitionofablackhole.Hehadnotrealizedthattheboundariesofthe
black hole according to the two definitions would be the same, and
hencesowouldtheirareas,providedtheblackholehadsettleddownto
astateinwhichitwasnotchangingwithtime.
FIGURE7.2ANDFIGURE7.3
The nondecreasing behavior of a black hole’s area was very
reminiscentofthebehaviorofaphysicalquantitycalledentropy,which
measuresthedegreeofdisorderofasystem.Itisamatterofcommon
experience that disorder will tend to increase if things are left to
themselves.(One has only to stop makingrepairsaround the house to
see that!) One can create order out of disorder (for example, one can
paintthehouse),butthatrequiresexpenditureofeffortorenergyandso
decreasestheamountoforderedenergyavailable.
A precise statement of this idea is known as the second law of
thermodynamics.Itstatesthattheentropyofanisolatedsystemalways
increases,andthatwhentwosystemsarejoinedtogether,theentropyof
the combined system is greater than the sum of the entropies of the
individualsystems.Forexample,considerasystemofgasmoleculesina
box.Themoleculescanbethoughtofaslittlebilliardballscontinually
colliding with each other and bouncing off the walls of the box. The
higherthetemperatureofthegas,thefasterthemoleculesmove,andso
the more frequently and harder they collide with the walls of the box
andthegreatertheoutwardpressure theyexerton thewalls.Suppose
thatinitiallythemoleculesareallconfinedtotheleft-handsideofthe
boxbyapartition.Ifthepartitionisthenremoved,themoleculeswill
tendtospreadoutandoccupybothhalvesofthebox.Atsomelatertime
theycould,bychance,allbeintherighthalforbackinthelefthalf,but
it is overwhelmingly more probable that there will be roughly equal
numbers in the two halves. Such a state is less ordered, or more
disordered,thantheoriginalstateinwhichallthemoleculeswereinone
half. One therefore says that the entropy of the gas has gone up.
Similarly, suppose one starts with two boxes, one containing oxygen
moleculesandtheothercontainingnitrogenmolecules.Ifonejoinsthe
boxes together and removes the intervening wall, the oxygen and the
nitrogenmoleculeswillstarttomix.Atalatertimethemostprobable
state would be a fairly uniform mixture of oxygen and nitrogen
moleculesthroughoutthe twoboxes.This statewould beless ordered,
and hence have more entropy, than the initial state of two separate
boxes.
Thesecondlawofthermodynamicshasaratherdifferentstatusthan
that of other laws of science, such as Newton’s law of gravity, for
example, because it does not hold always, just in the vast majority of
cases. The probability of all the gas molecules in our first box being
foundinonehalfoftheboxatalatertimeismanymillionsofmillions
toone,butitcanhappen.However,ifonehasablackholearound,there
seemstobearathereasierwayofviolatingthesecondlaw:justthrow
somematterwithalotofentropy,suchasaboxofgas,downtheblack
hole.Thetotalentropyofmatteroutsidetheblackholewouldgodown.
One could, of course, still say that the total entropy, including the
entropyinsidetheblackhole,hasnotgonedown—butsincethereisno
waytolookinsidetheblackhole,wecannotseehowmuchentropythe
matterinsideithas.Itwouldbenice,then,iftherewassomefeatureof
theblackholebywhichobserversoutside theblack holecould tellits
entropy, and which would increase whenever matter carrying entropy
fellintotheblackhole.Followingthediscovery,describedabove,that
theareaoftheeventhorizonincreasedwhenevermatterfellintoablack
hole,aresearchstudentatPrincetonnamedJacobBekensteinsuggested
thattheareaoftheeventhorizonwasameasureoftheentropyofthe
blackhole.Asmattercarryingentropyfellintoablackhole,theareaof
itseventhorizonwouldgoup,sothatthesumoftheentropyofmatter
outsideblackholesandtheareaofthehorizonswouldnevergodown.
Thissuggestionseemedtopreventthesecondlawofthermodynamics
from being violated in most situations. However, there was one fatal
flaw. If a black hole has entropy, then it ought also to have a
temperature. But a body with a particular temperature must emit
radiationatacertainrate.Itisa matterofcommonexperiencethatif
oneheatsupapokerinafireitglowsredhotandemitsradiation,but
bodies at lower temperatures emit radiation too; one just does not
normallynoticeit becausetheamount isfairlysmall. Thisradiationis
requiredinordertopreventviolationofthesecondlaw.Soblackholes
ought to emit radiation. But by their very definition, black holes are
objectsthatarenotsupposedtoemitanything.Itthereforeseemedthat
theareaoftheeventhorizonofablackholecouldnotberegardedasits
entropy.In1972IwroteapaperwithBrandonCarterandanAmerican
colleague, Jim Bardeen, in which we pointed out that although there
were many similarities between entropy and the area of the event
horizon,therewasthisapparentlyfataldifficulty.Imustadmitthatin
writingthispaperIwasmotivatedpartlybyirritationwithBekenstein,
who,Ifelt,hadmisusedmydiscoveryoftheincreaseoftheareaofthe
eventhorizon.However,itturnedoutintheendthathewasbasically
correct,thoughinamannerhehadcertainlynotexpected.
In September 1973, while I was visiting Moscow, I discussed black
holes with two leading Soviet experts, Yakov Zeldovich and Alexander
Starobinsky. They convinced me that, according to the quantum
mechanicaluncertaintyprinciple,rotatingblackholesshouldcreateand
emitparticles.Ibelievedtheirargumentsonphysicalgrounds,butIdid
notlikethemathematicalwayinwhichtheycalculatedtheemission.I
therefore set about devising a better mathematical treatment, which I
described at an informal seminar in Oxford at the end of November
1973.AtthattimeIhadnotdonethecalculationstofindouthowmuch
wouldactuallybeemitted.Iwasexpectingtodiscoverjusttheradiation
thatZeldovichandStarobinskyhadpredictedfromrotatingblackholes.
However, when I did the calculation, I found, to my surprise and
annoyance,thatevennon-rotatingblackholesshouldapparentlycreate
andemitparticlesatasteadyrate.AtfirstIthoughtthatthisemission
indicatedthatoneoftheapproximationsIhadusedwasnotvalid.Iwas
afraidthatifBekensteinfoundoutaboutit,hewoulduseitasafurther
argumenttosupporthisideasabouttheentropyofblackholes,whichI
still did not like. However, the more I thought about it, the more it
seemed thatthe approximations really ought to hold. But what finally
convincedmethat theemission wasreal wasthat thespectrum ofthe
emitted particles was exactly that which would be emitted by a hot
body, and that the black hole was emitting particles at exactly the
correct rate to prevent violations of the second law. Since then the
calculationshavebeenrepeatedinanumberofdifferentformsbyother
people.They all confirm that a black hole ought to emit particles and
radiationasifitwereahotbodywithatemperaturethatdependsonly
ontheblackhole’smass:thehigherthemass,thelowerthetemperature.
Howisitpossiblethatablackholeappearstoemitparticleswhenwe
know that nothing can escape from within its event horizon? The
answer,quantumtheorytellsus,isthattheparticlesdonotcomefrom
withintheblackhole,butfromthe“empty”spacejustoutsidetheblack
hole’seventhorizon!Wecanunderstandthisinthefollowingway:what
wethinkofas“empty”spacecannotbecompletelyemptybecausethat
would mean that all the fields, such as the gravitational and
electromagnetic fields, would have to be exactly zero. However, the
valueofafieldanditsrateofchangewithtimearelikethepositionand
velocity of a particle: the uncertainty principle implies that the more
accuratelyoneknowsoneofthesequantities,thelessaccuratelyonecan
knowtheother.Soinemptyspacethefieldcannotbefixedatexactly
zero, because then it would have both a precise value (zero) and a
precise rate of change (also zero). There must be a certain minimum
amountofuncertainty,orquantumfluctuations,inthevalueofthefield.
Onecanthinkofthesefluctuationsaspairsofparticlesoflightorgravity
thatappeartogetheratsometime,moveapart,andthencometogether
againandannihilateeachother.Theseparticlesarevirtualparticleslike
the particles that carry the gravitational force of the sun: unlike real
particles, they cannot be observed directly with a particle detector.
However,their indirecteffects, suchassmall changesin the energyof
electronorbitsinatoms,canbemeasuredandagreewiththetheoretical
predictions to a remarkable degree of accuracy. The uncertainty
principlealsopredictsthattherewillbesimilarvirtualpairsofmatter
particles,suchaselectronsorquarks.Inthiscase,however,onemember
of the pair will be a particle and the other an antiparticle (the
antiparticlesoflightandgravityarethesameastheparticles).
Becauseenergycannotbecreatedoutofnothing,oneofthepartners
in a particle/antiparticle pair will have positive energy, and the other
partnernegativeenergy.Theonewithnegativeenergyiscondemnedto
be a short-lived virtual particle because real particles always have
positive energy in normal situations. It must therefore seek out its
partner and annihilate with it. However, a real particle close to a
massivebodyhaslessenergythanifitwerefaraway,becauseitwould
takeenergytoliftitfarawayagainstthegravitationalattractionofthe
body. Normally, the energy of the particle is still positive, but the
gravitational field inside a black hole is so strong that even a real
particle can have negative energy there. It is therefore possible, if a
blackholeispresent,forthevirtualparticlewithnegativeenergytofall
intotheblackholeandbecomearealparticleorantiparticle.Inthiscase
itnolongerhastoannihilatewithitspartner.Itsforsakenpartnermay
fallintotheblackholeaswell.Or,havingpositiveenergy,itmightalso
escape from the vicinity of the black hole as a real particle or
antiparticle (Fig. 7.4). To an observer at a distance, it will appear to
havebeenemittedfromtheblackhole.Thesmallertheblackhole,the
shorter the distance the particle with negative energy will have to go
before it becomes a real particle, and thus the greater the rate of
emission,andtheapparenttemperature,oftheblackhole.
Thepositiveenergyoftheoutgoingradiationwouldbebalancedbya
flow of negative energy particles into the black hole. By Einstein’s
equationE=mc2(whereEisenergy,mismass,andcisthespeedof
light),energyisproportionaltomass.Aflowofnegativeenergyintothe
blackholethereforereducesitsmass.Astheblackholelosesmass,the
areaofitseventhorizongetssmaller,butthisdecreaseintheentropyof
the black hole is more than compensated for by the entropy of the
emittedradiation,sothesecondlawisneverviolated.
Moreover, the lower the mass of the black hole, the higher its
temperature.Soastheblackholelosesmass,itstemperatureandrateof
emissionincrease,soitlosesmassmorequickly.Whathappenswhenthe
massoftheblackholeeventuallybecomesextremelysmallisnotquite
clear, but the most reasonable guess is that it would disappear
completely in a tremendous final burst of emission, equivalent to the
explosionofmillionsofH-bombs.
FIGURE7.4
Ablack hole with a mass a few times that ofthesunwould have a
temperatureofonlyonetenmillionthofadegreeaboveabsolutezero.
Thisismuchlessthanthetemperatureofthemicrowaveradiationthat
fillstheuniverse (about2.7°above absolutezero),so suchblackholes
wouldemitevenlessthantheyabsorb.Iftheuniverseisdestinedtogo
onexpandingforever,thetemperatureofthemicrowaveradiationwill
eventually decrease to less than that of such a black hole, which will
thenbegintolosemass.But,eventhen,itstemperaturewouldbesolow
thatitwouldtakeaboutamillionmillionmillionmillionmillionmillion
millionmillionmillionmillionmillionyears(1withsixty-sixzerosafter
it) to evaporate completely. This is much longer than the age of the
universe,whichisonlyabouttenortwentythousandmillionyears(1or
2withtenzerosafterit).Ontheotherhand,asmentionedinChapter6,
there might be primordial black holes with a very much smaller mass
thatweremadebythecollapseofirregularitiesintheveryearlystages
oftheuniverse.Suchblackholeswouldhaveamuchhighertemperature
and would be emitting radiation at a much greater rate. A primordial
blackholewithaninitialmassofathousandmilliontonswouldhavea
lifetimeroughlyequaltotheageoftheuniverse.Primordialblackholes
withinitialmasseslessthanthisfigurewouldalreadyhavecompletely
evaporated, but those with slightly greater masses would still be
emittingradiationintheformofXraysandgammarays.TheseXrays
and gamma rays are like waves of light, but with a much shorter
wavelength.Suchholeshardlydeservetheepithetblack:theyreallyare
white hot and are emitting energy at a rate of about ten thousand
megawatts.
One such black hole could run ten large power stations, if only we
could harness its power. This would be rather difficult, however: the
blackholewouldhavethemassofamountaincompressedintolessthan
amillionmillionthofaninch,thesizeofthenucleusofanatom!Ifyou
hadoneoftheseblackholesonthesurfaceoftheearth,therewouldbe
no way to stop it from falling through the floor to the center of the
earth.Itwouldoscillatethroughtheearthandback,untileventuallyit
settleddownatthecenter.Sotheonlyplacetoputsuchablackhole,in
which one might use the energy that it emitted, would be in orbit
around the earth—and the only way that one could getit to orbit the
earthwouldbetoattractittherebytowingalargemassinfrontofit,
ratherlikeacarrotinfrontofadonkey.Thisdoesnotsoundlikeavery
practicalproposition,atleastnotintheimmediatefuture.
But even if we cannot harness the emission from these primordial
blackholes,whatareourchancesofobservingthem?Wecouldlookfor
the gamma rays that the primordial black holes emit during most of
their lifetime. Although the radiation from most would be very weak
because they are far away, the total from all of them might be
detectable.We do observesucha background of gammarays: Fig. 7.5
shows how the observed intensity differs at different frequencies (the
number of waves per second). However, this background could have
been, and probably was, generated by processes other than primordial
blackholes.ThedottedlineinFig.7.5showshowtheintensityshould
varywithfrequencyforgammaraysgivenoffbyprimordialblackholes,
iftherewereonaverage300percubiclight-year.Onecanthereforesay
thattheobservationsofthegammaraybackgrounddonotprovideany
positiveevidenceforprimordialblackholes,buttheydotellusthaton
averagetherecannotbemorethan300ineverycubiclight-yearinthe
universe.Thislimitmeansthatprimordialblackholescouldmakeupat
mostonemillionthofthematterintheuniverse.
With primordial black holes being so scarce, it might seem unlikely
thattherewouldbeonenearenoughforustoobserveasanindividual
source of gamma rays. But since gravity would draw primordial black
holes toward any matter, they should be much more common in and
around galaxies. So although the gamma ray background tells us that
therecanbenomorethan300primordialblack holespercubiclight-
yearonaverage,ittellsusnothingabouthowcommontheymightbein
ourowngalaxy. Ifthey were,say, amillion timesmore commonthan
this,thenthenearestblackholetouswouldprobablybeatadistanceof
aboutathousandmillionkilometers,oraboutasfarawayasPluto,the
farthestknownplanet.Atthisdistanceitwouldstillbeverydifficultto
detectthesteadyemissionofablackhole,evenifitwastenthousand
megawatts.Inordertoobserveaprimordialblackholeonewouldhave
to detect several gamma ray quanta coming from the same direction
withinareasonablespaceoftime,suchasaweek.Otherwise,theymight
simplybepartofthebackground.ButPlanck’squantumprincipletells
us that each gamma ray quantum has a very high energy, because
gamma rays have a very high frequency, so it would not take many
quanta to radiate even ten thousand megawatts. And to observe these
fewcomingfromthedistanceofPlutowouldrequirealargergammaray
detector than any that have been constructed so far. Moreover, the
detector would have to be in space, because gamma rays cannot
penetratetheatmosphere.
FIGURE7.5
Ofcourse,ifablackholeascloseasPlutoweretoreachtheendofits
lifeandblowup,itwouldbeeasytodetectthefinalburstofemission.
But if the black hole has been emitting for the last ten or twenty
thousand million years, the chance of it reaching the end of its life
withinthenextfewyears,ratherthanseveralmillionyearsinthepastor
future,isreallyrathersmall!Soinordertohaveareasonablechanceof
seeinganexplosionbeforeyourresearchgrantranout,youwouldhave
to find a way to detect any explosions within a distance of about one
light-year.Infactburstsofgammaraysfromspacehavebeendetected
bysatellitesoriginallyconstructedtolookforviolationsoftheTestBan
Treaty. These seem to occur about sixteen times a month and to be
roughlyuniformlydistributedindirectionacrossthesky.Thisindicates
thattheycomefromoutsidetheSolarSystemsinceotherwisewewould
expect them to be concentrated toward the plane of the orbits of the
planets. The uniform distribution also indicates that the sources are
eitherfairlyneartousinourgalaxyorrightoutsideitatcosmological
distancesbecauseotherwise,again,theywouldbeconcentratedtoward
theplaneofthegalaxy.Inthelattercase,theenergyrequiredtoaccount
fortheburstswouldbefartoohightohavebeenproducedbytinyblack
holes,butifthesourceswerecloseingalacticterms,itmightbepossible
thattheywereexplodingblackholes.Iwouldverymuchlikethistobe
the case but I have to recognize that there are other possible
explanationsforthegammaraybursts,suchascollidingneutronstars.
New observations in the next few years, particularly by gravitational
wavedetectorslikeLIGO,shouldenableustodiscovertheoriginofthe
gammaraybursts.
Even if the search for primordial black holes proves negative, as it
seemsitmay,itwillstillgiveusimportantinformationaboutthevery
early stages of the universe. If the early universe had been chaotic or
irregular, or if the pressure of matter had been low, one would have
expectedittoproducemanymoreprimordialblackholesthanthelimit
alreadysetby ourobservationsof thegammaray background.Onlyif
theearlyuniversewasverysmoothanduniform,withahighpressure,
canoneexplaintheabsenceofobservablenumbersofprimordialblack
holes.
The idea of radiation from black holes was the first example of a
predictionthatdependedinanessentialwayonboththegreattheories
ofthiscentury,generalrelativityandquantummechanics.Itarouseda
lotofoppositioninitiallybecauseitupsettheexistingviewpoint:“How
canablackholeemitanything?”WhenIfirstannouncedtheresultsof
mycalculations ata conferenceat the Rutherford-AppletonLaboratory
near Oxford, I was greeted with general incredulity. At the end of my
talk the chairman of the session, John G. Taylor from Kings College,
London, claimed it was all nonsense. He even wrote a paper to that
effect. However, in the end most people, including John Taylor, have
cometotheconclusionthatblackholesmustradiatelikehotbodiesif
our other ideas about general relativity and quantum mechanics are
correct. Thus, even though we have not yet managed to find a
primordialblackhole,thereisfairlygeneralagreementthatifwedid,it
wouldhavetobeemittingalotofgammaraysandXrays.
The existence of radiation from black holes seems to imply that
gravitationalcollapseisnotasfinalandirreversibleasweoncethought.
If an astronaut falls into a black hole, its mass will increase, but
eventuallytheenergyequivalentofthatextramasswillbereturnedto
theuniverseintheformofradiation.Thus,inasense,theastronautwill
be“recycled.”Itwouldbeapoorsortofimmortality,however,because
any personal concept of time for the astronaut would almost certainly
come to an end as he was torn apart inside the black hole! Even the
typesofparticlesthatwereeventuallyemittedbytheblackholewould
ingeneralbedifferentfromthosethatmadeuptheastronaut:theonly
featureoftheastronautthatwouldsurvivewouldbehismassorenergy.
The approximations I used to derive the emission from black holes
shouldworkwellwhentheblackholehasamassgreaterthanafraction
ofagram.However,theywillbreakdownattheendoftheblackhole’s
lifewhenitsmassgetsverysmall.Themostlikelyoutcomeseemstobe
that the black hole will just disappear, at least from ourregion of the
universe,takingwithittheastronautandanysingularitytheremightbe
inside it, if indeed there is one. This was the first indication that
quantummechanicsmightremovethesingularitiesthatwerepredicted
bygeneralrelativity.However,themethodsthatIandotherpeoplewere
using in 1974 were not able to answer questions such as whether
singularities would occur in quantum gravity. From 1975 onward I
therefore started to develop a more powerful approach to quantum
gravity based on Richard Feynman’s idea of a sum over histories. The
answers that this approach suggests for the origin and fate of the
universe and its contents, such as astronauts, will be described in the
nexttwochapters.Weshallseethatalthoughtheuncertaintyprinciple
placeslimitations on theaccuracy of allour predictions, itmay at the
same time remove the fundamental unpredictability that occurs at a
space-timesingularity.
E
CHAPTER8
THEORIGINAND
FATEOFTHE
UNIVERSE
instein’sgeneraltheoryofrelativity,onitsown,predictedthatspace-
time began at the big bang singularity and would come to an end
eitheratthebigcrunchsingularity(ifthewholeuniverserecollapsed),
oratasingularityinsideablackhole(ifalocalregion,suchasastar,
weretocollapse).Anymatterthatfellintotheholewouldbedestroyed
at the singularity, and only the gravitational effect of its mass would
continue to be felt outside. On the other hand, when quantum effects
weretakenintoaccount,itseemedthatthemassorenergyofthematter
would eventually be returned to the rest of the universe, and that the
blackhole,alongwithanysingularityinsideit,would evaporateaway
and finally disappear. Could quantum mechanics have an equally
dramaticeffectonthebigbangandbigcrunchsingularities?Whatreally
happens during the very early or late stages of the universe, when
gravitational fields are so strong that quantum effects cannot be
ignored?Doestheuniverseinfacthaveabeginningoranend?Andif
so,whataretheylike?
Throughoutthe1970sIhadbeenmainlystudyingblackholes,butin
1981myinterestinquestionsabouttheoriginandfateoftheuniverse
wasreawakenedwhenIattendedaconferenceoncosmologyorganized
by the Jesuits in the Vatican. The Catholic Church had made a bad
mistakewithGalileowhenittriedtolaydownthelawonaquestionof
science, declaring that the sun went round the earth. Now, centuries
later, it had decided to invite a number of experts to advise it on
cosmology.Atthe endofthe conferencethe participantswere granted
anaudiencewiththePope.Hetoldusthatitwasallrighttostudythe
evolutionoftheuniverseafterthebigbang,butweshouldnotinquire
into the big bang itself because that was the moment of Creation and
therefore the work of God. I was glad then that he did not know the
subjectofthetalkIhadjustgivenattheconference—thepossibilitythat
space-timewasfinitebuthadnoboundary,whichmeansthatithadno
beginning,nomomentofCreation.Ihadnodesiretosharethefateof
Galileo,withwhomIfeelastrongsenseofidentity,partlybecauseofthe
coincidenceofhavingbeenbornexactly300yearsafterhisdeath!
InordertoexplaintheideasthatIandotherpeoplehavehadabout
howquantummechanicsmayaffecttheoriginandfateoftheuniverse,
itisnecessaryfirsttounderstandthegenerallyacceptedhistoryofthe
universe,accordingtowhatisknownasthe“hotbigbangmodel.”This
assumesthattheuniverseisdescribedbyaFriedmannmodel,rightback
tothebigbang.Insuchmodelsonefindsthatastheuniverseexpands,
anymatterorradiationinitgetscooler.(Whentheuniversedoublesin
size,itstemperaturefallsbyhalf.)Sincetemperatureissimplyameasure
of the average energy—or speed—of the particles, this cooling of the
universe would have a major effect on the matter in it. At very high
temperatures,particleswouldbemovingaroundsofastthattheycould
escape any attraction toward each other due to nuclear or
electromagneticforces,butastheycooledoffonewouldexpectparticles
that attract each other to start to clump together. Moreover, even the
types of particles that exist in the universe would depend on the
temperature. At high enough temperatures, particles have so much
energy that whenever they collide many different particle/antiparticle
pairswould be produced—andalthough someof these particles would
annihilateonhittingantiparticles,theywouldbeproducedmorerapidly
than they could annihilate. At lower temperatures, however, when
collidingparticleshavelessenergy,particle/antiparticlepairswouldbe
produced less quickly—and annihilation would become faster than
production.
At the big bang itself the universe is thought to havehad zero size,
and so to have been infinitely hot. But as the universe expanded, the
temperatureoftheradiationdecreased.Onesecondafterthebigbang,it
wouldhavefallentoabouttenthousandmilliondegrees.Thisisabouta
thousand times the temperature at the center of the sun, but
temperaturesashighasthisarereachedinH-bombexplosions.Atthis
timetheuniversewouldhavecontainedmostlyphotons,electrons,and
neutrinos (extremely light particles that are affected only by the weak
force and gravity) and their antiparticles, together with some protons
andneutrons.Astheuniversecontinuedtoexpandandthetemperature
to drop, the rate at which electron/antielectron pairs were being
producedin collisions would have fallenbelowthe rate at which they
were being destroyed by annihilation. So most of the electrons and
antielectronswouldhaveannihilatedwitheach otherto producemore
photons, leaving only a few electrons left over. The neutrinos and
antineutrinos, however, would not have annihilated with each other,
becausetheseparticlesinteractwiththemselvesandwithotherparticles
only very weakly. So they should still be around today. If we could
observethem,itwouldprovideagoodtestofthispictureofaveryhot
early stage of the universe. Unfortunately, their energies nowadays
wouldbetoolowforustoobservethemdirectly.However,ifneutrinos
are not massless, but have a small mass of their own,as suggested by
some recent experiments, we might be able to detect them indirectly:
theycouldbeaformof“darkmatter,”likethatmentionedearlier,with
sufficientgravitational attractionto stop theexpansion of the universe
andcauseittocollapseagain.
Aboutonehundredsecondsafterthebigbang,thetemperaturewould
havefallentoonethousandmilliondegrees,thetemperatureinsidethe
hotteststars.Atthistemperatureprotonsandneutronswouldnolonger
have sufficient energy to escape the attraction of the strong nuclear
force,andwouldhavestartedtocombinetogethertoproducethenuclei
ofatomsofdeuterium(heavyhydrogen),whichcontainoneprotonand
one neutron. The deuterium nuclei would then have combined with
more protons and neutrons to make helium nuclei, which contain two
protonsandtwoneutrons,andalsosmallamountsofacoupleofheavier
elements,lithium andberyllium. Onecancalculate thatin the hotbig
bang model about a quarter of the protons and neutrons would have
beenconvertedintoheliumnuclei,alongwithasmallamountofheavy
hydrogen and other elements. The remaining neutrons would have
decayedintoprotons,whicharethenucleiofordinaryhydrogenatoms.
Thispictureofahotearlystageoftheuniversewasfirstputforward
bythescientistGeorgeGamowinafamouspaperwrittenin1948witha
student of his, Ralph Alpher. Gamow had quite a sense of humor—he
persuadedthenuclearscientistHansBethetoaddhisnametothepaper
tomakethelistofauthors“Alpher,Bethe,Gamow,”likethefirstthree
letters of the Greek alphabet, alpha, beta, gamma: particularly
appropriateforapaperonthebeginningoftheuniverse!Inthispaper
they made the remarkable prediction that radiation (in the form of
photons)from the very hot early stages of the universe should still be
around today, but with its temperature reduced to only a few degrees
above absolute zero (—273°C). It was this radiation that Penzias and
Wilsonfoundin1965.AtthetimethatAlpher,Bethe,andGamowwrote
theirpaper,notmuchwasknownaboutthenuclearreactionsofprotons
andneutrons.Predictionsmadefortheproportionsofvariouselements
in the early universe were therefore rather inaccurate, but these
calculations have been repeated in the light of better knowledge and
nowagreeverywellwithwhatweobserve.Itis,moreover,verydifficult
toexplaininanyotherwaywhythereshouldbesomuchheliuminthe
universe. We are therefore fairly confident that we have the right
picture,atleastbacktoaboutonesecondafterthebigbang.
Withinonlyafewhoursofthebigbang,theproductionofheliumand
otherelementswouldhavestopped.Andafterthat,forthenextmillion
yearsorso,theuniversewouldhavejustcontinuedexpanding,without
anything much happening. Eventually, once the temperature had
droppedtoafewthousanddegrees,andelectronsandnucleinolonger
hadenoughenergytoovercometheelectromagneticattractionbetween
them,theywouldhavestartedcombiningtoformatoms.Theuniverseas
a whole would have continued expanding and cooling, but in regions
thatwereslightlydenserthanaverage,theexpansionwouldhavebeen
sloweddownbytheextragravitationalattraction.Thiswouldeventually
stopexpansioninsomeregionsandcausethemtostarttorecollapse.As
they were collapsing, the gravitational pull of matter outside these
regionsmightstartthemrotatingslightly.Asthe collapsingregiongot
smaller,itwouldspinfaster—justasskatersspinningonicespinfaster
as they draw in their arms. Eventually, when the region got small
enough, it would be spinning fast enough to balance the attraction of
gravity, and in this way disklike rotating galaxies were born. Other
regions,whichdidnothappentopickuparotation,wouldbecomeoval-
shapedobjectscalledellipticalgalaxies.Inthese,theregionwouldstop
collapsing because individual parts of the galaxy would be orbiting
stablyrounditscenter,butthegalaxywouldhavenooverallrotation.
Astimewenton,thehydrogenandheliumgasinthegalaxieswould
break up into smaller clouds that would collapse under their own
gravity. As these contracted, and the atoms within them collided with
oneanother,thetemperatureofthegaswouldincrease,untileventually
it became hot enough to start nuclear fusion reactions. These would
convert the hydrogen into more helium, and the heat given off would
raisethepressure,andsostopthecloudsfromcontractinganyfurther.
Theywouldremainstableinthisstateforalongtimeasstarslikeour
sun,burninghydrogenintoheliumandradiatingtheresultingenergyas
heatandlight. Moremassive starswould needto behotter tobalance
their stronger gravitational attraction, making the nuclear fusion
reactions proceed so much more rapidly that they would use up their
hydrogen in as little as a hundred million years. They would then
contractslightly,andastheyheatedupfurther,wouldstarttoconvert
helium into heavier elements like carbon or oxygen. This, however,
would not release much more energy, so a crisis would occur, as was
described in the chapter on black holes. What happens next is not
completelyclear,butitseemslikelythatthecentralregionsofthestar
would collapse to a very dense state, such as a neutron star or black
hole. The outer regions of the star may sometimes get blown off in a
tremendousexplosioncalledasupernova,whichwouldoutshineallthe
otherstarsinitsgalaxy.Someoftheheavierelementsproducednearthe
endofthestar’slifewouldbeflungbackintothegasinthegalaxy,and
wouldprovidesomeoftherawmaterialforthenextgenerationofstars.
Our own sun contains about 2 percent of these heavier elements,
because it is a second-or third-generation star, formed some five
thousandmillionyearsagooutofacloudofrotatinggascontainingthe
debrisofearliersupernovas.Mostofthegasinthatcloudwenttoform
thesunorgotblownaway,butasmallamountoftheheavierelements
collectedtogethertoformthebodiesthatnoworbitthesunasplanets
liketheearth.
The earth was initially very hot and without an atmosphere. In the
courseoftimeitcooledandacquiredanatmospherefromtheemission
ofgasesfromtherocks.Thisearlyatmospherewasnotoneinwhichwe
couldhavesurvived.Itcontainednooxygen,butalotofothergasesthat
arepoisonoustous,suchashydrogensulfide(thegasthatgivesrotten
eggstheirsmell).Thereare,however,otherprimitiveformsoflifethat
canflourishundersuchconditions.Itisthoughtthattheydevelopedin
the oceans, possibly as a result of chance combinations of atoms into
large structures, called macromolecules, which were capable of
assemblingotheratomsintheoceanintosimilarstructures.Theywould
thus have reproduced themselves and multiplied. In some cases there
would be errors in the reproduction. Mostly these errors would have
been such that the new macromolecule could not reproduce itself and
eventually would have been destroyed. However, a few of the errors
would have produced new macromolecules that were even better at
reproducing themselves. They would have therefore had an advantage
andwouldhavetendedtoreplacetheoriginalmacromolecules.Inthis
waya process of evolution was started that led to the development of
more and more complicated, self-reproducing organisms. The first
primitiveformsoflifeconsumedvariousmaterials,includinghydrogen
sulfide,andreleasedoxygen.Thisgraduallychangedtheatmosphereto
the composition that it has today, and allowed the development of
higherformsoflifesuchasfish,reptiles,mammals,andultimatelythe
humanrace.
This picture of a universe that started off very hot and cooled as it
expanded is in agreement with all the observational evidence that we
have today. Nevertheless, it leaves a number of important questions
unanswered:
1. Whywastheearlyuniversesohot?
2. Whyistheuniversesouniformonalargescale?Whydoesitlook
thesameatallpointsofspaceandinalldirections?Inparticular,
whyisthetemperatureofthemicrowavebackgroundradiationso
nearlythesamewhenwelookindifferentdirections?Itisabitlike
asking a number of students an exam question. If they all give
exactly the same answer, you can be pretty sure they have
communicatedwitheachother.Yet,inthemodeldescribedabove,
therewouldnothavebeentimesincethebigbangforlighttoget
from one distant region to another, even though the regions were
close together in the early universe. According to the theory of
relativity,iflight cannotgetfrom oneregionto another,noother
information can. So there would be no way in which different
regions in the early universe could have come to have the same
temperatureaseachother,unlessforsomeunexplainedreasonthey
happenedtostartoutwiththesametemperature.
3. Why did the universe start out with so nearly the critical rate of
expansionthatseparatesmodelsthatrecollapsefromthosethatgo
on expanding forever, that even now, ten thousand million years
later, it is still expanding at nearly the critical rate? If the rate of
expansiononesecondafterthebigbanghadbeensmallerbyeven
onepartinahundredthousandmillionmillion,theuniversewould
haverecollapsedbeforeiteverreacheditspresentsize.
4. Despitethefactthattheuniverseissouniformandhomogeneouson
a large scale, it contains local irregularities, such as stars and
galaxies. These are thought to have developed from small
differencesinthedensityoftheearlyuniversefromoneregionto
another.Whatwastheoriginofthesedensityfluctuations?
The general theory of relativity, on its own, cannot explain these
features or answer these questions because of its prediction that the
universestartedoffwithinfinitedensityatthebigbangsingularity.At
the singularity, general relativity and all other physical laws would
break down: one couldn’t predict what would come out of the
singularity.Asexplainedbefore,thismeansthatonemightaswellcut
thebigbang,andanyeventsbeforeit,outofthetheory,becausethey
can have no effect on what we observe. Space-time would have a
boundary—abeginningatthebigbang.
Scienceseemstohaveuncoveredasetoflawsthat,withinthelimits
set by the uncertainty principle, tell us how the universe will develop
withtime, ifweknow itsstateat anyone time. Theselaws may have
originallybeendecreedbyGod,butitappearsthathehassinceleftthe
universetoevolveaccordingtothemanddoesnotnowinterveneinit.
Buthowdidhechoosetheinitialstateorconfigurationoftheuniverse?
Whatwerethe“boundaryconditions”atthebeginningoftime?
OnepossibleansweristosaythatGodchosetheinitialconfiguration
of the universe for reasons that we cannot hope to understand. This
wouldcertainlyhavebeenwithinthepowerofanomnipotentbeing,but
if he had started it off in such an incomprehensible way, why did he
choosetoletitevolveaccordingtolawsthatwecouldunderstand?The
wholehistoryofsciencehasbeenthegradualrealizationthateventsdo
not happen in an arbitrary manner, but that they reflect a certain
underlyingorder,whichmayormaynotbedivinelyinspired.Itwould
beonlynaturaltosupposethatthisordershouldapplynotonlytothe
laws, but also to the conditions at the boundary of space-time that
specifytheinitialstateoftheuniverse.Theremaybealargenumberof
modelsoftheuniversewithdifferentinitialconditionsthatallobeythe
laws.Thereoughttobesomeprinciplethatpicksoutoneinitialstate,
andhenceonemodel,torepresentouruniverse.
One such possibility is what are called chaotic boundary conditions.
These implicitly assume either that the universe is spatially infinite or
that there are infinitely many universes. Under chaotic boundary
conditions, the probability of finding any particular region of space in
any given configuration just after the big bang is the same, in some
sense, as the probability of finding it in any other configuration: the
initialstateoftheuniverseischosenpurelyrandomly.Thiswouldmean
that the early universe would have probably been very chaotic and
irregular because there are many more chaotic and disordered
configurationsfortheuniversethantherearesmoothandorderedones.
(Ifeachconfigurationisequallyprobable,itislikely thatthe universe
startedoutinachaoticanddisorderedstate,simplybecausethereareso
many more of them.) It is difficult to see how such chaotic initial
conditions could have given rise to a universe that is so smooth and
regularonalargescaleasoursistoday.Onewouldalsohaveexpected
thedensityfluctuationsinsuchamodeltohaveledtotheformationof
manymoreprimordialblackholesthantheupperlimitthathasbeenset
byobservationsofthegammaraybackground.
If the universe is indeed spatially infinite, or if there are infinitely
manyuniverses,therewouldprobablybesomelargeregionssomewhere
thatstartedoutinasmoothanduniformmanner.Itisabitlikethewell-
known horde of monkeys hammering away on typewriters—most of
whattheywritewillbegarbage,butveryoccasionallybypurechance
theywilltypeoutoneofShakespeare’ssonnets.Similarly,inthecaseof
theuniverse,coulditbethatwearelivinginaregionthatjusthappens
bychancetobesmoothanduniform?Atfirstsightthismightseemvery
improbable, because such smooth regions would be heavily
outnumbered by chaotic and irregular regions. However, suppose that
only in the smooth regions were galaxies and stars formed and were
conditions right for the development of complicated self-replicating
organismslikeourselveswhowerecapableofaskingthequestion:why
istheuniversesosmooth?Thisisanexampleoftheapplicationofwhat
isknownastheanthropicprinciple,whichcanbeparaphrasedas“We
seetheuniversethewayitisbecauseweexist.”
There are two versions of the anthropic principle, the weak and the
strong. The weak anthropic principle states that in a universe that is
largeorinfinite inspaceand/or time,theconditions necessaryforthe
developmentof intelligent life will be metonlyin certain regions that
are limited in space and time. The intelligent beings in these regions
shouldthereforenotbesurprisediftheyobservethattheirlocalityinthe
universesatisfiestheconditionsthatarenecessaryfortheirexistence.It
isabit likearich personlivingin awealthyneighborhoodnotseeing
anypoverty.
Oneexampleoftheuseoftheweakanthropicprincipleisto“explain”
why the big bang occurred about ten thousand million years ago—it
takes about that long for intelligent beings to evolve. As explained
above, an early generation of stars first had to form. These stars
convertedsomeoftheoriginalhydrogenandheliumintoelementslike
carbonandoxygen,outofwhichwearemade.Thestarsthenexploded
as supernovas, and their debris went to form other stars and planets,
among them those of our Solar System, which is about five thousand
million years old. The first one or two thousand million years of the
earth’s existence were too hot for the development of anything
complicated. The remaining three thousand million years or so have
beentakenupbytheslowprocessofbiologicalevolution,whichhasled
from the simplest organisms to beings who are capable of measuring
timebacktothebigbang.
Few people would quarrel with the validity or utility of the weak
anthropic principle. Some, however, go much further and propose a
strongversionoftheprinciple.Accordingtothistheory,thereareeither
manydifferentuniversesormanydifferentregionsofasingleuniverse,
eachwithitsowninitialconfigurationand,perhaps,withitsownsetof
lawsofscience.Inmostoftheseuniversestheconditionswouldnotbe
right for the development of complicated organisms; only in the few
universesthatarelikeourswouldintelligentbeingsdevelopandaskthe
question,“Whyistheuniversethewayweseeit?”Theansweristhen
simple:ifithadbeendifferent,wewouldnotbehere!
The laws of science, as we know them at present, contain many
fundamentalnumbers,likethesizeoftheelectricchargeoftheelectron
andtheratioofthemassesoftheprotonandtheelectron.Wecannot,at
themomentatleast,predictthevaluesofthesenumbersfromtheory—
wehavetofindthembyobservation.Itmaybethatonedayweshall
discoveracompleteunifiedtheorythatpredictsthemall,butitisalso
possible that some or all of them vary from universe to universe or
withinasingleuniverse.Theremarkablefactisthatthevaluesofthese
numbers seem to have been very finely adjusted to make possible the
developmentof life.For example,if the electriccharge of theelectron
hadbeenonlyslightlydifferent,starseitherwouldhavebeenunableto
burn hydrogen and helium, or else they would not have exploded. Of
course, there might be other forms of intelligent life, not dreamed of
evenbywritersofsciencefiction,thatdidnotrequirethelightofastar
likethesunortheheavierchemicalelementsthataremadeinstarsand
areflungbackintospacewhenthestarsexplode.Nevertheless,itseems
clearthattherearerelativelyfewrangesofvaluesforthenumbersthat
wouldallowthedevelopmentofanyformofintelligentlife.Mostsetsof
values would give rise to universes that, although they might be very
beautiful,wouldcontainnooneabletowonderatthatbeauty.Onecan
take this either as evidence of a divine purpose in Creation and the
choice of the laws of science or as support for the strong anthropic
principle.
There are a number of objections that one can raise to the strong
anthropic principle as an explanation of the observed state of the
universe.First,inwhatsensecanallthesedifferentuniversesbesaidto
exist? If they are really separate from each other, what happens in
another universe can have no observable consequences in our own
universe. We should therefore use the principle of economy and cut
them out of the theory. If, on the other hand, they are just different
regions of a single universe, the laws of science would have to be the
same in each region, because otherwise one could not move
continuouslyfromoneregiontoanother.Inthiscasetheonlydifference
between the regions would be their initial configurations and so the
stronganthropicprinciplewouldreducetotheweakone.
A second objection to the strong anthropic principle is that it runs
againstthetideofthewholehistoryofscience.Wehavedevelopedfrom
the geocentric cosmologies of Ptolemy and his forebears, through the
heliocentriccosmologyofCopernicusandGalileo,tothemodernpicture
inwhichtheearthisamedium-sizedplanetorbitingaroundanaverage
starintheoutersuburbsofanordinaryspiralgalaxy,whichisitselfonly
oneofabout amillionmillion galaxiesintheobservableuniverse.Yet
the strong anthropic principle would claim that this whole vast
constructionexistssimplyforoursake.Thisisveryhardtobelieve.Our
SolarSystemiscertainlyaprerequisiteforourexistence,andonemight
extendthistothewholeofourgalaxytoallowforanearliergeneration
ofstarsthatcreatedtheheavierelements.Buttheredoesnotseemtobe
any need for all those other galaxies, nor for the universe to be so
uniformandsimilarineverydirectiononthelargescale.
One would feel happier about the anthropic principle, at least in its
weakversion,ifonecouldshowthatquiteanumberofdifferentinitial
configurations for the universe would have evolved to produce a
universe like the one we observe. If this is the case, a universe that
developedfromsomesortofrandominitialconditionsshouldcontaina
numberofregionsthataresmoothanduniformandaresuitableforthe
evolutionofintelligentlife.Ontheotherhand,iftheinitialstateofthe
universehadtobechosenextremelycarefullytoleadtosomethinglike
whatweseearoundus,theuniversewouldbeunlikelytocontainany
regioninwhichlifewouldappear.Inthehotbigbangmodeldescribed
above,therewasnotenoughtimeintheearlyuniverseforheattohave
flowedfromoneregiontoanother.Thismeansthattheinitialstateof
the universe would have to have had exactly the same temperature
everywhere in order to account for the fact that the microwave
background has the same temperature in every direction we look. The
initialrateofexpansionalsowouldhavehadtobechosenveryprecisely
fortherateofexpansionstilltobesoclosetothecriticalrateneededto
avoidrecollapse.Thismeansthattheinitialstateoftheuniversemust
havebeenverycarefullychosenindeedifthehotbigbangmodelwas
correctrightbacktothebeginningoftime.Itwouldbeverydifficultto
explainwhytheuniverseshouldhavebeguninjustthisway,exceptas
theactofaGodwhointendedtocreatebeingslikeus.
Inanattempttofindamodeloftheuniverseinwhichmanydifferent
initialconfigurationscouldhaveevolvedto somethinglikethe present
universe,a scientistatthe Massachusetts Instituteof Technology,Alan
Guth, suggested that the early universe might have gone through a
period of very rapid expansion. This expansion is said to be
“inflationary,” meaning that the universe at one time expanded at an
increasing rate rather than the decreasing rate that it does today.
According to Guth, the radius of the universe increased by a million
millionmillionmillionmillion(1withthirtyzerosafterit)timesinonly
atinyfractionofasecond.
Guth suggested that the universe started out from the big bang in a
veryhot,butratherchaotic,state.Thesehightemperatureswouldhave
meantthattheparticlesintheuniversewouldbemovingveryfastand
would have high energies. As we discussed earlier, one would expect
thatatsuchhightemperaturesthestrongandweaknuclearforcesand
theelectromagneticforcewouldallbeunifiedintoasingleforce.Asthe
universeexpanded,itwouldcool,andparticleenergieswouldgodown.
Eventually there would be what is called a phase transition and the
symmetrybetweentheforceswouldbebroken:thestrongforcewould
become different from the weak and electromagnetic forces. One
commonexampleofaphasetransitionisthefreezingofwaterwhenyou
coolitdown.Liquidwaterissymmetrical,thesameateverypointand
in every direction. However, when ice crystals form, they will have
definite positions and will be lined up in some direction. This breaks
water’ssymmetry.
Inthecaseofwater,ifoneiscareful,onecan“supercool”it:thatis,
onecanreducethetemperaturebelowthefreezingpoint(0°C)without
iceforming.Guthsuggestedthattheuniversemightbehaveinasimilar
way: the temperature might drop below the critical value without the
symmetry between the forces being broken. If this happened, the
universe would be in an unstable state, with more energy than if the
symmetryhadbeenbroken.Thisspecialextraenergycanbeshownto
have an antigravitational effect: it would have acted just like the
cosmological constant that Einstein introduced into general relativity
whenhewastryingtoconstructastaticmodeloftheuniverse.Sincethe
universewouldalreadybeexpandingjustasinthehotbigbangmodel,
therepulsiveeffect ofthiscosmologicalconstantwouldthereforehave
made the universe expand at an ever-increasing rate. Even in regions
wherethereweremorematterparticlesthanaverage,thegravitational
attractionofthematterwouldhavebeenoutweighedbytherepulsionof
the effective cosmological constant. Thus these regions would also
expand in an accelerating inflationary manner. As they expanded and
the matter particles got farther apart, one would be left with an
expandinguniversethatcontainedhardlyanyparticlesandwasstillin
the supercooled state. Any irregularities in the universe would simply
havebeensmoothedoutbytheexpansion,asthewrinklesinaballoon
aresmoothedawaywhenyoublowitup.Thusthepresentsmoothand
uniform state of the universe could have evolved from many different
non-uniforminitialstates.
In such a universe, in which the expansion was accelerated by a
cosmological constant rather than slowed down by the gravitational
attractionofmatter,therewouldbeenoughtimeforlighttotravelfrom
oneregiontoanotherintheearlyuniverse.Thiscouldprovideasolution
to the problem, raised earlier, of why different regions in the early
universe have the same properties. Moreover, the rate of expansion of
theuniversewouldautomaticallybecomeveryclosetothecriticalrate
determined by the energy density of the universe. This could then
explain why the rate of expansion is still so close to the critical rate,
without having to assume that the initial rate of expansion of the
universewasverycarefullychosen.
Theideaofinflationcouldalsoexplainwhythereissomuchmatterin
the universe. There are something like ten million million million
million million million million million million million million million
millionmillion(1witheightyzerosafterit)particlesintheregionofthe
universe that we can observe. Where did they all come from? The
answeristhat,inquantumtheory,particlescanbecreatedoutofenergy
intheformofparticle/antiparticlepairs.Butthatjustraisesthequestion
ofwheretheenergycamefrom.Theansweristhatthetotalenergyof
theuniverseisexactlyzero.Thematterintheuniverseismadeoutof
positive energy. However, the matter is all attracting itself by gravity.
Twopiecesofmatterthatareclosetoeachotherhavelessenergythan
the same two pieces a long way apart, because you have to expend
energy to separate them against the gravitational force that is pulling
them together. Thus, in a sense, the gravitational field has negative
energy.Inthecaseofauniversethatisapproximatelyuniforminspace,
onecanshowthatthisnegativegravitationalenergyexactlycancelsthe
positive energy represented by the matter. So the total energy of the
universeiszero.
Nowtwicezeroisalsozero.Thustheuniversecandoubletheamount
of positive matter energy and also double the negative gravitational
energy without violation of the conservation of energy. This does not
happen in the normal expansion of the universe in which the matter
energy density goes down as the universe gets bigger. It does happen,
however,intheinflationaryexpansionbecausetheenergydensityofthe
supercooled state remains constant while the universe expands: when
theuniversedoublesinsize,thepositivematterenergyandthenegative
gravitational energy both double, so the total energy remains zero.
Duringthe inflationaryphase, theuniverseincreases itssize byavery
large amount. Thus the total amount of energy available to make
particles becomes very large. As Guth has remarked, “It is said that
there’snosuchthingasafreelunch.Buttheuniverseistheultimatefree
lunch.”
Theuniverseisnotexpandinginaninflationarywaytoday.Thusthere
hastobesomemechanismthatwouldeliminatetheverylargeeffective
cosmological constant and so change the rate of expansion from an
acceleratedonetoonethatissloweddownbygravity,aswehavetoday.
In the inflationary expansion one might expect that eventually the
symmetry between the forces would be broken, just as supercooled
water always freezes in the end. The extra energy of the unbroken
symmetrystatewouldthenbereleasedandwouldreheattheuniverseto
atemperaturejustbelowthecriticaltemperatureforsymmetrybetween
theforces.Theuniversewouldthengoontoexpandandcooljustlike
thehotbigbangmodel,buttherewouldnowbeanexplanationofwhy
theuniversewasexpandingatexactlythecriticalrateandwhydifferent
regionshadthesametemperature.
InGuth’soriginalproposalthephasetransitionwassupposedtooccur
suddenly,ratherliketheappearanceoficecrystalsinverycoldwater.
The idea was that “bubbles” of the new phase of broken symmetry
wouldhaveformedintheoldphase,likebubblesofsteamsurrounded
by boiling water. The bubbles were supposed to expand and meet up
with each other until the whole universe was in the new phase. The
troublewas,asIandseveralotherpeoplepointedout,thattheuniverse
wasexpandingsofastthatevenifthebubblesgrewatthespeedoflight,
theywouldbemovingawayfromeachotherandsocouldnotjoinup.
The universe would be left in a very non-uniform state, with some
regionsstillhavingsymmetrybetweenthedifferentforces.Suchamodel
oftheuniversewouldnotcorrespondtowhatwesee.
In October 1981, I went to Moscow for a conference on quantum
gravity.AftertheconferenceIgaveaseminarontheinflationarymodel
andits problems at theSternbergAstronomical Institute. Beforethis, I
hadgotsomeoneelsetogivemylecturesforme,becausemostpeople
couldnotunderstandmyvoice.Buttherewasnottimetopreparethis
seminar,soIgaveitmyself,withoneofmygraduatestudentsrepeating
my words. It worked well, and gave me much more contact with my
audience.IntheaudiencewasayoungRussian,AndreiLinde,fromthe
LebedevInstituteinMoscow.Hesaidthatthedifficultywiththebubbles
not joining up could be avoided if the bubbles were so big that our
regionoftheuniverseisallcontainedinsideasinglebubble.Inorderfor
thistowork,thechangefromsymmetrytobrokensymmetrymusthave
taken place very slowly inside the bubble, but this is quite possible
accordingto grandunified theories.Linde’s ideaofa slowbreaking of
symmetry was very good, but I later realized that his bubbles would
have to have been bigger than the size of the universe at the time! I
showedthatinsteadthesymmetrywouldhavebrokeneverywhereatthe
sametime,ratherthanjustinsidebubbles.Thiswouldleadtoauniform
universe,asweobserve.Iwasveryexcitedbythisideaanddiscussedit
withoneofmystudents,IanMoss.AsafriendofLinde’s,Iwasrather
embarrassed, however, when I was later sent his paper by a scientific
journalandaskedwhetheritwassuitableforpublication.Irepliedthat
therewasthisflawaboutthebubblesbeingbiggerthantheuniverse,but
that the basic idea of a slow breaking of symmetry was very good. I
recommended that the paper be published as it was because it would
take Linde several months to correct it, since anything he sent to the
WestwouldhavetobepassedbySovietcensorship,whichwasneither
very skillful nor very quick with scientific papers. Instead, I wrote a
shortpaperwithIanMossinthesamejournalinwhichwepointedout
thisproblemwiththebubbleandshowedhowitcouldberesolved.
ThedayafterIgotbackfromMoscowIsetoutforPhiladelphia,where
IwasduetoreceiveamedalfromtheFranklinInstitute.Mysecretary,
Judy Fella, had used her not inconsiderable charm to persuade British
AirwaystogiveherselfandmefreeseatsonaConcordeasapublicity
venture.However,Iwashelduponmywaytotheairportbyheavyrain
andImissedtheplane.Nevertheless,IgottoPhiladelphiaintheendand
received my medal. I was then asked to give a seminar on the
inflationary universe at Drexel University in Philadelphia. I gave the
sameseminarabouttheproblemsoftheinflationaryuniverse,justasin
Moscow.
A very similar idea to Linde’s was put forth independently a few
monthslaterbyPaulSteinhardtandAndreasAlbrechtoftheUniversity
ofPennsylvania.TheyarenowgivenjointcreditwithLindeforwhatis
called “the new inflationary model,” based on the idea of a slow
breakingofsymmetry.(TheoldinflationarymodelwasGuth’soriginal
suggestionoffastsymmetrybreakingwiththeformationofbubbles.)
The new inflationary model was a good attempt to explain why the
universe is the way it is. However, I and several other people showed
that,atleastinitsoriginalform,itpredictedmuchgreatervariationsin
the temperature of the microwave background radiation than are
observed. Later work has also cast doubt on whether there could be a
phasetransitionintheveryearlyuniverseofthekindrequired.Inmy
personalopinion,thenewinflationarymodelisnowdeadasascientific
theory,althoughalotofpeopledonotseemtohaveheardofitsdemise
andarestillwritingpapersasifitwereviable.Abettermodel,calledthe
chaotic inflationary model, was put forward by Linde in 1983. In this
there is no phase transition or supercooling. Instead, there is a spin 0
field,which,becauseofquantumfluctuations,wouldhavelargevalues
insome regionsof theearly universe.The energyofthe fieldin those
regions would behave like a cosmological constant. It would have a
repulsivegravitationaleffect,andthusmakethoseregionsexpandinan
inflationarymanner.Astheyexpanded,theenergyofthefieldinthem
would slowly decrease until the inflationary expansion changed to an
expansion like that in the hot big bang model. One of these regions
wouldbecomewhatwenowseeastheobservableuniverse.Thismodel
hasalltheadvantagesoftheearlierinflationarymodels,butitdoesnot
depend on a dubious phase transition, and it can moreover give a
reasonablesizeforthefluctuationsinthetemperatureofthemicrowave
backgroundthatagreeswithobservation.
Thisworkoninflationarymodelsshowedthatthepresentstateofthe
universecouldhavearisenfromquitealargenumberofdifferentinitial
configurations.Thisisimportant,becauseitshowsthattheinitialstate
of the part of the universe that we inhabit did not have to be chosen
withgreatcare.Sowemay,ifwewish,usetheweakanthropicprinciple
toexplainwhytheuniverselooksthewayitdoesnow.Itcannotbethe
case, however, that every initial configuration would have led to a
universe like the one we observe. One can show this by considering a
very different state for the universe at the present time, say, a very
lumpyandirregularone.Onecouldusethelawsofsciencetoevolvethe
universe back in time to determine its configuration at earlier times.
According to the singularity theorems of classical general relativity,
therewouldstillhavebeenabigbangsingularity.Ifyouevolvesucha
universeforwardintimeaccordingtothelawsofscience,youwillend
upwiththelumpyandirregularstateyoustartedwith.Thustheremust
have been initial configurations that would not have given rise to a
universeliketheoneweseetoday.Soeventheinflationarymodeldoes
not tell us why the initial configuration was not such as to produce
something very different from what we observe. Must we turn to the
anthropicprincipleforanexplanation?Wasit alljust aluckychance?
That would seem a counsel of despair, a negation of all our hopes of
understandingtheunderlyingorderoftheuniverse.
Inordertopredicthowtheuniverseshouldhavestartedoff,oneneeds
lawsthatholdatthebeginningoftime.Iftheclassicaltheoryofgeneral
relativitywascorrect,thesingularitytheoremsthatRogerPenroseandI
proved show that the beginning of time would have been a point of
infinitedensityandinfinitecurvatureofspace-time.Alltheknownlaws
ofscience would breakdown at sucha point. Onemightsuppose that
there were new laws that held at singularities, but it would be very
difficulteventoformulatesuchlawsatsuchbadlybehavedpoints,and
wewouldhavenoguidefromobservationsastowhatthoselawsmight
be. However, what the singularity theorems really indicate is that the
gravitationalfieldbecomessostrongthatquantumgravitationaleffects
becomeimportant:classicaltheoryisnolongeragooddescriptionofthe
universe.Soonehastouseaquantumtheoryofgravitytodiscussthe
very early stages of the universe. As we shall see, it is possible in the
quantum theory for the ordinary laws of science to hold everywhere,
includingatthebeginningoftime:itisnotnecessarytopostulatenew
lawsforsingularities,becausethereneednotbeanysingularitiesinthe
quantumtheory.
We don’t yet have a complete and consistent theory that combines
quantummechanicsandgravity.However,wearefairlycertainofsome
features that such a unified theory should have. One is that it should
incorporateFeynman’sproposaltoformulatequantumtheoryintermsof
a sum over histories. In this approach, a particle does not have just a
singlehistory,asitwouldinaclassicaltheory.Instead,itissupposedto
followeverypossiblepathinspace-time,andwitheachofthesehistories
thereareassociatedacoupleofnumbers,onerepresentingthesizeofa
waveandtheotherrepresentingitspositioninthecycle(itsphase).The
probabilitythattheparticle,say,passesthroughsomeparticularpointis
foundbyaddingupthewavesassociatedwitheverypossiblehistorythat
passes through that point. When one actually tries to perform these
sums,however,onerunsintoseveretechnicalproblems.Theonlyway
aroundtheseisthefollowingpeculiarprescription:onemustaddupthe
wavesforparticlehistoriesthatarenotinthe“real”timethatyouandI
experience but take place in what is called imaginary time. Imaginary
time may sound like science fiction but it is in fact a well-defined
mathematicalconcept.Ifwetake anyordinary(or“real”) numberand
multiply it by itself, the result is a positive number. (For example, 2
times2is4,butsois−2times−2.)Thereare,however,specialnumbers
(calledimaginarynumbers)thatgivenegativenumberswhenmultiplied
bythemselves.(Theonecalledi,whenmultipliedbyitself,gives−1,2i
multipliedbyitselfgives−4,andsoon.)
One can picture real and imaginary numbers in the following way:
Therealnumberscanberepresentedbyalinegoingfromlefttoright,
withzerointhemiddle,negativenumberslike−1,−2,etc.ontheleft,
andpositivenumbers, 1,2, etc. onthe right. Thenimaginarynumbers
are represented by a line going up and down the page, with i, 2i, etc.
abovethemiddle,and−i,−2i,etc.below.Thusimaginarynumbersare
inasensenumbersatrightanglestoordinaryrealnumbers.
ToavoidthetechnicaldifficultieswithFeynman’ssumoverhistories,
one must use imaginary time. That is to say, for the purposes of the
calculation one must measure time using imaginary numbers, rather
than real ones. This has an interesting effect on space-time: the
distinctionbetweentimeandspacedisappearscompletely.Aspace-time
inwhicheventshaveimaginaryvaluesofthetimecoordinateissaidto
beEuclidean,aftertheancientGreekEuclid,whofoundedthestudyof
thegeometryoftwo-dimensionalsurfaces.WhatwenowcallEuclidean
space-timeisverysimilarexceptthatithasfourdimensionsinsteadof
two. In Euclidean space-time there is no difference between the time
directionanddirectionsinspace.Ontheotherhand,inrealspace-time,
in which events are labeled by ordinary, real values of the time
coordinate, it is easy to tell the difference—the time direction at all
pointslieswithinthelightcone,andspacedirectionslieoutside.Inany
case, as far as everyday quantum mechanics is concerned, we may
regardouruseofimaginarytimeandEuclideanspace-timeasmerelya
mathematical device (or trick) to calculate answers about real space-
time.
Asecondfeaturethatwebelievemustbepartofanyultimatetheory
is Einstein’s idea that the gravitational field is represented by curved
space-time:particlestrytofollowthenearestthingtoastraightpathin
acurvedspace,butbecausespace-timeisnotflattheirpathsappearto
be bent, as if by a gravitational field. When we apply Feynman’s sum
overhistoriestoEinstein’sviewofgravity,theanalogueofthehistoryof
a particle is now a complete curved space-time that represents the
history of the whole universe. To avoid the technical difficulties in
actually performing the sum over histories, these curved space-times
must be taken to be Euclidean. That is, time is imaginary and is
indistinguishablefromdirectionsinspace.Tocalculatetheprobabilityof
findingarealspace-timewithsomecertainproperty,suchaslookingthe
same at every point and in every direction, one adds up the waves
associatedwithallthehistoriesthathavethatproperty.
Inthe classical theory ofgeneralrelativity, there aremanydifferent
possible curved space-times, each corresponding to a different initial
state of the universe. If we knew the initial state of our universe, we
would know its entire history. Similarly, in the quantum theory of
gravity, there are many different possible quantum states for the
universe.Again,ifweknewhowtheEuclideancurvedspace-timesinthe
sumoverhistoriesbehavedatearlytimes,wewouldknowthequantum
stateoftheuniverse.
Inthe classicaltheory of gravity,which isbased on realspace-time,
thereareonlytwopossiblewaystheuniversecanbehave:eitherithas
existedforaninfinitetime,orelseithadabeginningatasingularityat
some finite time in the past. In the quantum theory of gravity, on the
other hand, a third possibility arises. Because one is using Euclidean
space-times, in which the time direction is on the same footing as
directionsinspace,itispossibleforspace-timetobefiniteinextentand
yettohavenosingularitiesthatformedaboundaryoredge.Space-time
wouldbelikethesurfaceoftheearth,onlywithtwomoredimensions.
Thesurfaceoftheearthisfiniteinextentbutitdoesn’thaveaboundary
oredge:ifyousailoffintothesunset,youdon’tfallofftheedgeorrun
intoasingularity.(Iknow,becauseIhavebeenroundtheworld!)
If Euclidean space-time stretches back to infinite imaginary time, or
elsestartsatasingularityinimaginarytime,wehavethesameproblem
asintheclassicaltheory ofspecifying theinitial stateof theuniverse:
God may know how the universe began, but we cannot give any
particularreasonforthinkingitbeganonewayratherthananother.On
the other hand, the quantum theory of gravity has opened up a new
possibility,inwhichtherewouldbenoboundarytospace-timeandso
therewouldbenoneedtospecifythebehaviorattheboundary.There
wouldbenosingularitiesatwhichthelawsofsciencebrokedown,and
no edge of space-time at which one would have to appeal to God or
somenewlawtosettheboundaryconditionsforspace-time.Onecould
say:“Theboundaryconditionoftheuniverseisthatithasnoboundary.”
The universe would be completely self-contained and not affected by
anything outside itself. It would neither be created nor destroyed. It
wouldjustBE.
Itwasatthe conferenceintheVaticanmentionedearlierthatIfirst
putforwardthesuggestionthatmaybetimeandspacetogetherformeda
surfacethatwasfiniteinsize butdidnot haveanyboundary oredge.
Mypaperwasrathermathematical,however,soitsimplicationsforthe
roleofGodinthecreationoftheuniversewerenotgenerallyrecognized
atthetime(justaswellforme).AtthetimeoftheVaticanconference,I
did not know how to use the “no boundary” idea to make predictions
about the universe. However, I spent the following summer at the
UniversityofCalifornia,SantaBarbara.Thereafriendandcolleagueof
mine, Jim Hartle, worked out with me what conditions the universe
must satisfy if space-time had no boundary. When I returned to
Cambridge, I continued this work with two of my research students,
JulianLuttrelandJonathanHalliwell.
I’dliketoemphasizethatthisideathattimeandspaceshouldbefinite
“withoutboundary”isjustaproposal:itcannotbededucedfromsome
otherprinciple.Likeanyotherscientifictheory,itmayinitiallybeput
forward for aesthetic or metaphysical reasons, but the real test is
whetheritmakespredictionsthatagreewithobservation.This,however,
isdifficulttodetermineinthecaseofquantumgravity,fortworeasons.
First, as will be explained in Chapter 11, we are not yet sure exactly
which theory successfully combines general relativity and quantum
mechanics, though we know quite a lot about the form such a theory
must have. Second, any model that described the whole universe in
detailwouldbemuchtoocomplicatedmathematicallyforustobeable
to calculate exact predictions. One therefore has to make simplifying
assumptions and approximations—and even then, the problem of
extractingpredictionsremainsaformidableone.
Eachhistoryinthesumoverhistorieswilldescribenotonlythespace-
timebuteverythinginitaswell,includinganycomplicatedorganisms
likehumanbeingswhocanobservethehistoryoftheuniverse.Thismay
provide another justification for the anthropic principle, for if all the
historiesarepossible,thensolongasweexistinoneofthehistories,we
mayusetheanthropicprincipletoexplainwhytheuniverseisfoundto
be the way it is. Exactly what meaning can be attached to the other
histories,inwhichwedonotexist,isnotclear.Thisviewofaquantum
theory of gravity would be much more satisfactory, however, if one
couldshow that, using the sum over histories, our universe is not just
oneofthepossiblehistoriesbutoneofthemostprobableones.Todo
this,wemustperformthesumoverhistoriesforallpossibleEuclidean
space-timesthathavenoboundary.
Underthe“noboundary”proposalonelearnsthatthechanceofthe
universe being found to be following most of the possible histories is
negligible, but there is a particular family of histories that are much
moreprobablethantheothers.Thesehistoriesmaybepicturedasbeing
like the surface of the earth, with the distance from the North Pole
representingimaginarytimeandthesizeofacircleofconstantdistance
from the North Pole representing the spatial size of the universe. The
universestartsattheNorthPoleasasinglepoint.Asonemovessouth,
the circles of latitude at constant distance from the North Pole get
bigger, corresponding to the universe expanding with imaginary time
(Fig.8.1).Theuniversewouldreachamaximumsizeattheequatorand
wouldcontractwithincreasingimaginarytimetoasinglepointatthe
SouthPole.EventhoughtheuniversewouldhavezerosizeattheNorth
andSouthPoles,thesepointswouldnotbesingularities,anymorethan
theNorthandSouthPolesontheeartharesingular.Thelawsofscience
willholdatthem,justastheydoattheNorthandSouthPolesonthe
earth.
The history of the universe in real time, however, would look very
different.At about ten or twenty thousandmillion years ago, it would
havea minimum size,which was equaltothe maximum radiusof the
historyinimaginarytime.Atlaterrealtimes,theuniversewouldexpand
like the chaotic inflationary model proposed by Linde (but one would
notnowhavetoassumethattheuniversewascreatedsomehowinthe
rightsortofstate).Theuniversewouldexpandtoaverylargesize(Fig.
8.1) and eventually it would collapse again into what looks like a
singularityinrealtime.Thus,inasense,wearestillalldoomed,evenif
wekeepawayfromblackholes.Onlyifwecouldpicturetheuniversein
termsofimaginarytimewouldtherebenosingularities.
Ifthe universe really isin such a quantumstate,there would be no
singularities in the history of the universe in imaginary time. It might
seem therefore that my more recent work had completely undone the
resultsofmyearlierworkonsingularities.But,asindicatedabove,the
realimportanceofthesingularitytheoremswasthatthey showedthat
thegravitationalfieldmustbecomesostrongthatquantumgravitational
effects could not be ignored. This in turn led to the idea that the
universe could be finite in imaginary time but without boundaries or
singularities. When one goes back to the real time in which we live,
however, there will still appear to be singularities. The poor astronaut
whofallsintoablackholewillstillcometoastickyend;onlyifhelived
inimaginarytimewouldheencounternosingularities.
FIGURE8.1
Thismightsuggestthattheso-calledimaginarytimeisreallythereal
time, and that what we call real time is just a figment of our
imaginations.Inrealtime,theuniversehasabeginningandanendat
singularitiesthatformaboundarytospace-timeandatwhichthelaws
ofsciencebreakdown.Butinimaginarytime,therearenosingularities
or boundaries. So maybe what we call imaginary time is really more
basic, and what we call real is just an idea that we invent to help us
describe what we think the universe is like. But according to the
approach I described in Chapter 1, a scientific theory is just a
mathematicalmodelwemaketodescribeourobservations:itexistsonly
in our minds. So it is meaningless to ask: which is real, “real” or
“imaginary” time? It is simply a matter of which is the more useful
description.
Onecanalsousethesumoverhistories,alongwiththenoboundary
proposal, to find which properties of the universe are likely to occur
together. For example, one can calculate the probability that the
universeisexpandingatnearlythesamerateinalldifferentdirections
atatimewhenthedensityoftheuniversehasitspresentvalue.Inthe
simplifiedmodelsthathavebeenexaminedsofar,thisprobabilityturns
outtobehigh;thatis,theproposednoboundaryconditionleadstothe
prediction that it is extremely probable that the present rate of
expansionoftheuniverseis almostthesameineachdirection.Thisis
consistentwiththeobservationsofthemicrowavebackgroundradiation,
which show that it has almost exactly the same intensity in any
direction.Iftheuniversewereexpandingfasterinsomedirectionsthan
in others, the intensity of the radiation in those directions would be
reducedbyanadditionalredshift.
Furtherpredictionsofthenoboundaryconditionarecurrentlybeing
worked out. A particularly interesting problem is the size of the small
departures from uniform density in the early universe that caused the
formation first of the galaxies, then of stars, and finally of us. The
uncertainty principle implies that the early universe cannot have been
completelyuniformbecausetheremusthavebeensomeuncertaintiesor
fluctuationsinthepositionsandvelocitiesoftheparticles.Usingtheno
boundarycondition,wefindthattheuniversemustinfacthavestarted
off with just the minimum possible non-uniformity allowed by the
uncertaintyprinciple.Theuniversewouldhavethenundergoneaperiod
ofrapidexpansion,asintheinflationarymodels.Duringthisperiod,the
initialnon-uniformitieswouldhavebeenamplified untiltheywerebig
enoughtoexplaintheoriginofthestructuresweobservearoundus.In
1992 the Cosmic Background Explorer satellite (COBE) first detected
veryslightvariationsintheintensityofthemicrowavebackgroundwith
direction.Thewaythesenon-uniformitiesdependondirectionseemsto
agree with the predictions of the inflationary model and the no
boundaryproposal.Thusthenoboundaryproposalisagoodscientific
theory in the sense of Karl Popper: it could have been falsified by
observations but instead its predictions have been confirmed. In an
expandinguniverseinwhichthedensityofmattervariedslightlyfrom
place to place, gravity would have caused the denser regions to slow
down their expansion and start contracting. This would lead to the
formationofgalaxies,stars,andeventuallyeveninsignificantcreatures
like ourselves. Thus all the complicated structures that we see in the
universe might be explained by the no boundary condition for the
universetogetherwiththeuncertaintyprincipleofquantummechanics.
The idea that space and time may form a closed surface without
boundary also has profound implications for the role of God in the
affairs of the universe. With the success of scientific theories in
describingevents,mostpeoplehavecometobelievethatGodallowsthe
universetoevolveaccordingtoasetoflawsanddoesnotintervenein
theuniversetobreaktheselaws.However,thelawsdonottelluswhat
theuniverseshouldhavelookedlikewhenitstarted—itwouldstillbe
uptoGodtowinduptheclockworkandchoosehowtostartitoff.So
longastheuniversehadabeginning,wecouldsupposeithadacreator.
But if the universe is really completely self-contained, having no
boundary or edge, it would have neither beginning nor end: it would
simplybe.Whatplace,then,foracreator?
I
CHAPTER9
THEARROW
OFTIME
npreviouschapterswehaveseenhowourviewsofthenatureoftime
have changed over the years. Up to the beginning of this century
peoplebelievedinanabsolutetime.Thatis,eacheventcouldbelabeled
byanumbercalled“time”inauniqueway,andallgoodclockswould
agreeonthetimeintervalbetweentwoevents.However,thediscovery
thatthespeedoflightappearedthesametoeveryobserver,nomatter
howhewasmoving,ledtothetheoryofrelativity—andinthatonehad
toabandontheideathattherewasauniqueabsolutetime.Instead,each
observer would have his own measure of time as recorded by a clock
that he carried: clocks carried by different observers would not
necessarilyagree.Thustimebecameamorepersonalconcept,relativeto
theobserverwhomeasuredit.
Whenonetriedtounifygravitywithquantummechanics,onehadto
introduce the idea of “imaginary” time. Imaginary time is
indistinguishablefromdirectionsinspace.Ifonecangonorth,onecan
turnaroundandheadsouth;equally,ifonecangoforwardinimaginary
time,oneoughttobeabletoturnroundandgobackward.Thismeans
that there can be no important difference between the forward and
backward directions of imaginary time. On the other hand, when one
looksat “real”time, there’s avery bigdifferencebetween theforward
and backward directions, as we all know. Where does this difference
betweenthepastandthefuturecomefrom?Whydowerememberthe
pastbutnotthefuture?
The laws of science do not distinguish between the past and the
future. More precisely, as explained earlier, the laws of science are
unchangedunderthecombinationofoperations(orsymmetries)known
as C, P, and T. (C means changing particles for antiparticles. P means
takingthemirrorimage,soleftandrightareinterchanged.AndTmeans
reversingthedirectionofmotionofallparticles:ineffect,runningthe
motion backward.) The laws of science that govern the behavior of
matterunderallnormalsituationsareunchangedunderthecombination
ofthetwooperationsCandPontheirown.Inotherwords,lifewould
be just the same for the inhabitants of another planet who were both
mirror images of us and who were made of antimatter, rather than
matter.
Ifthelawsofscienceareunchangedbythecombinationofoperations
C and P, and also by the combination C, P, and T, they must also be
unchanged under the operation T alone. Yet there is a big difference
between the forward and backward directions of real time in ordinary
life.Imagineacupofwaterfallingoffatableandbreakingintopieces
onthefloor.Ifyoutakeafilmofthis,youcaneasilytellwhetheritis
beingrunforwardorbackward.Ifyourunitbackwardyouwillseethe
piecessuddenlygatherthemselvestogetheroffthefloorandjumpback
toformawholecuponthetable.Youcantellthatthefilmisbeingrun
backward because this kind of behavior is never observed in ordinary
life.Ifitwere,crockerymanufacturerswouldgooutofbusiness.
Theexplanationthatisusuallygivenastowhywedon’tseebroken
cupsgatheringthemselvestogetheroffthefloorandjumpingbackonto
thetableis that it is forbidden by the second law of thermodynamics.
This says that in any closed system disorder, or entropy, always
increaseswithtime.Inotherwords,itisaformofMurphy’slaw:things
alwaystendtogowrong!Anintactcuponthetableisastateofhigh
order,buta brokencup onthe floorisadisorderedstate. Onecan go
readilyfromthecuponthetableinthepasttothebrokencuponthe
floorinthefuture,butnottheotherwayround.
Theincreaseofdisorderorentropywithtimeisoneexampleofwhat
iscalledanarrowoftime,somethingthatdistinguishesthepastfromthe
future, giving a direction to time. There are at least three different
arrows of time. First, there is the thermodynamic arrow of time, the
directionoftimeinwhichdisorderorentropyincreases.Then,thereis
thepsychologicalarrowoftime.Thisisthedirectioninwhichwefeel
time passes, the direction in which we remember the past but not the
future. Finally, there is the cosmological arrow of time. This is the
direction of time in which the universe is expanding rather than
contracting.
In this chapter I shall argue that the no boundary condition for the
universe, together with the weak anthropic principle, can explain why
allthreearrowspointinthesamedirection—andmoreover,whyawell-
defined arrow of time should exist at all. I shall argue that the
psychological arrow is determined by the thermodynamic arrow, and
thatthesetwoarrowsnecessarilyalwayspointinthesamedirection.If
one assumes the no boundary condition for the universe, we shall see
thattheremustbewell-definedthermodynamicandcosmologicalarrows
of time, but they will not point in the same direction for the whole
historyoftheuniverse.However,Ishallarguethatitisonlywhenthey
do point in the same direction that conditions are suitable for the
development of intelligent beings who can ask the question: why does
disorder increase in the same direction of time as that in which the
universeexpands?
Ishalldiscussfirstthethermodynamicarrowoftime.Thesecondlaw
of thermodynamics results from the fact that there are always many
more disordered states than there are ordered ones. For example,
consider the pieces of a jigsaw in a box. There is one, and only one,
arrangementinwhichthepiecesmakeacompletepicture.Ontheother
hand,thereareaverylargenumberofarrangementsinwhichthepieces
aredisorderedanddon’tmakeapicture.
Suppose a system starts out in one of the small number of ordered
states.Astimegoesby,thesystemwillevolveaccordingtothelawsof
scienceanditsstatewillchange.Atalatertime,itismoreprobablethat
thesystemwillbeinadisorderedstatethaninanorderedonebecause
there are more disordered states. Thus disorder will tend to increase
withtimeifthesystemobeysaninitialconditionofhighorder.
Suppose the pieces of the jigsaw start off in a box in the ordered
arrangement in which they form a picture. If you shake the box, the
pieces will take up another arrangement. This will probably be a
disorderedarrangementinwhichthepiecesdon’tformaproperpicture,
simplybecausetherearesomanymoredisorderedarrangements.Some
groups of pieces may still form parts of the picture, but the more you
shakethebox,themorelikelyitisthatthesegroupswillgetbrokenup
andthepieceswillbeinacompletelyjumbledstateinwhichtheydon’t
form any sort of picture. So the disorder of the pieces will probably
increasewithtimeifthepiecesobeytheinitialconditionthattheystart
offinaconditionofhighorder.
Suppose,however,thatGoddecidedthattheuniverseshouldfinishup
inastateofhighorderbutthatitdidn’tmatterwhatstateitstartedin.
Atearlytimestheuniversewouldprobablybeinadisorderedstate.This
would mean that disorder would decrease with time. You would see
broken cups gathering themselves together and jumping back onto the
table.However,anyhumanbeingswhowereobservingthecupswould
be living in a universe in which disorder decreased with time. I shall
argue that such beings would have a psychological arrow of time that
wasbackward.Thatis,theywouldremembereventsinthefuture,and
not remember events in their past. When the cup was broken, they
would remember it being on the table, but when it was on the table,
theywouldnotrememberitbeingonthefloor.
It is rather difficult to talk about human memory because we don’t
know how the brain works in detail. We do, however, know all about
how computer memories work. I shall therefore discuss the
psychological arrow of time for computers. I think it is reasonable to
assumethatthearrowforcomputersisthesameasthatforhumans.Ifit
werenot,onecouldmakeakillingonthestockexchangebyhavinga
computerthatwouldremembertomorrow’sprices!Acomputermemory
isbasicallyadevicecontainingelementsthatcanexistineitheroftwo
states.Asimpleexampleisanabacus.Initssimplestform,thisconsists
ofanumberofwires;oneachwirethereareanumberofbeadsthatcan
be put in one of two positions. Before an item is recorded in a
computer’s memory, the memory is in a disordered state, with equal
probabilitiesforthetwopossiblestates.(Theabacusbeadsarescattered
randomlyonthewiresoftheabacus.)Afterthememoryinteractswith
the system to be remembered, it will definitely be in one state or the
other,accordingtothestateofthesystem.(Eachabacusbeadwillbeat
eithertheleftortherightoftheabacuswire.)Sothememoryhaspassed
fromadisorderedstateto anordered one.However,inorderto make
surethatthememoryisintherightstate,itisnecessarytouseacertain
amount of energy (to move the bead or to power the computer, for
example).Thisenergyisdissipatedasheat,andincreasestheamountof
disorderintheuniverse.Onecanshowthatthisincreaseindisorderis
alwaysgreaterthantheincreaseintheorderofthememoryitself.Thus
the heat expelled by the computer’s cooling fan means that when a
computerrecordsaniteminmemory,thetotalamountofdisorderinthe
universe still goes up. The direction of time in which a computer
remembersthepastisthesameasthatinwhichdisorderincreases.
Oursubjectivesenseofthedirectionoftime,thepsychologicalarrow
oftime,isthereforedeterminedwithinourbrainbythethermodynamic
arrow of time. Just like a computer, we must remember things in the
order in which entropy increases. This makes the second law of
thermodynamicsalmosttrivial.Disorderincreaseswithtimebecausewe
measure time in the direction in which disorder increases. You can’t
haveasaferbetthanthat!
Butwhyshouldthethermodynamicarrowoftimeexistatall?Or,in
otherwords,whyshouldtheuniversebeinastateofhighorderatone
endof time,theend thatwecall thepast?Why isitnot in astate of
completedisorderatalltimes?Afterall,thismightseemmoreprobable.
Andwhyisthedirectionoftimeinwhichdisorderincreasesthesameas
thatinwhichtheuniverseexpands?
Intheclassicaltheoryofgeneralrelativityonecannotpredicthowthe
universewouldhavebegunbecausealltheknownlawsofsciencewould
havebrokendownatthebigbangsingularity.Theuniversecouldhave
startedoutinaverysmoothandorderedstate.Thiswouldhaveledto
well-defined thermodynamic and cosmological arrows of time, as we
observe.Butitcouldequallywellhavestartedoutinaverylumpyand
disorderedstate.Inthatcase,theuniversewouldalreadybeinastateof
complete disorder, so disorder could not increase with time. It would
either stay constant, in which case there would be no well-defined
thermodynamicarrowoftime,oritwoulddecrease,inwhichcasethe
thermodynamicarrowoftimewouldpointintheoppositedirectionto
thecosmologicalarrow. Neitherofthese possibilitiesagrees withwhat
we observe. However, as we have seen, classical general relativity
predicts its own downfall. When the curvature of space-time becomes
large, quantum gravitational effects will become important and the
classicaltheorywillceasetobeagooddescriptionoftheuniverse.One
hastouseaquantumtheoryofgravitytounderstandhowtheuniverse
began.
Inaquantumtheoryofgravity,aswesawinthelastchapter,inorder
tospecifythestateoftheuniverseonewouldstillhavetosayhowthe
possible histories of the universe would behave at the boundary of
space-time in the past. One could avoid this difficulty of having to
describewhatwedonotandcannotknowonlyifthehistoriessatisfythe
noboundarycondition:theyarefiniteinextentbuthavenoboundaries,
edges, or singularities. In that case, the beginning of time would be a
regular,smoothpointofspace-timeandtheuniversewouldhavebegun
itsexpansioninaverysmoothandorderedstate.Itcouldnothavebeen
completelyuniform,becausethatwouldviolatetheuncertaintyprinciple
ofquantumtheory.Therehadtobesmallfluctuationsinthedensityand
velocitiesofparticles.Thenoboundarycondition,however,impliedthat
these fluctuations were as small as they could be, consistent with the
uncertaintyprinciple.
Theuniverse wouldhave startedoff with aperiod of exponentialor
“inflationary”expansioninwhichitwouldhaveincreaseditssizebya
verylargefactor.Duringthisexpansion,thedensityfluctuationswould
have remained small at first, but later would have started to grow.
Regions in which the density was slightly higher than average would
havehadtheirexpansionsloweddownbythegravitationalattractionof
the extra mass. Eventually, such regions would stop expanding and
collapsetoformgalaxies,stars,andbeingslikeus.Theuniversewould
havestarted in a smooth and orderedstate, and would become lumpy
anddisorderedastimewenton.Thiswouldexplaintheexistenceofthe
thermodynamicarrowoftime.
Butwhatwouldhappenifandwhentheuniversestoppedexpanding
and began to contract? Would the thermodynamic arrow reverse and
disorder begin to decrease with time? This would lead to all sorts of
science-fiction-like possibilities for people who survived from the
expanding to the contracting phase. Would they see broken cups
gatheringthemselvestogether offthe floorandjumpingback ontothe
table?Wouldtheybeabletoremembertomorrow’spricesandmakea
fortune on the stock market? It might seem a bit academic to worry
aboutwhatwillhappenwhentheuniversecollapsesagain,asitwillnot
start to contract for at least another ten thousand million years. But
thereisaquickerwaytofindoutwhatwillhappen:jumpintoablack
hole.Thecollapseofastartoformablackholeisratherlikethelater
stages of the collapse of the whole universe. So if disorder were to
decreaseinthecontractingphaseoftheuniverse,onemightalsoexpect
ittodecreaseinsideablackhole.Soperhapsanastronautwhofellintoa
black hole would be able to make money at roulette by remembering
wheretheballwentbeforeheplacedhisbet.(Unfortunately,however,
hewouldnothavelongtoplaybeforehewasturnedtospaghetti.Nor
would he be able to let us know about the reversal of the
thermodynamicarrow,orevenbankhiswinnings,becausehewouldbe
trappedbehindtheeventhorizonoftheblackhole.)
At first, I believed that disorder would decrease when the universe
recollapsed.ThiswasbecauseIthoughtthattheuniversehadtoreturn
toasmoothandorderedstatewhenitbecamesmallagain.Thiswould
meanthat the contractingphase would belike the time reverseof the
expandingphase.Peopleinthecontractingphasewouldlivetheirlives
backward:theywoulddiebeforetheywerebornandgetyoungerasthe
universecontracted.
This idea is attractive because it would mean a nice symmetry
between the expanding and contracting phases. However, one cannot
adoptitonitsown,independentofotherideasabouttheuniverse.The
question is: is it implied by the no boundary condition, or is it
inconsistentwiththatcondition?AsIsaid,Ithoughtatfirstthattheno
boundary condition did indeed imply that disorder would decrease in
the contracting phase. I was misled partly by the analogy with the
surface of the earth. If one took the beginning of the universe to
correspond to the North Pole, then the end of the universe should be
similartothebeginning,justastheSouthPoleissimilartotheNorth.
However, the North and South Poles correspond to the beginning and
end of the universe in imaginary time. The beginning and end in real
timecanbeverydifferentfromeachother.IwasalsomisledbyworkI
had done on a simple model of the universe in which the collapsing
phaselookedlikethetimereverseoftheexpandingphase.However,a
colleagueofmine,DonPage,ofPennStateUniversity,pointedoutthat
the no boundary condition did not require the contracting phase
necessarilytobethetimereverseoftheexpandingphase.Further,one
of my students, Raymond Laflamme, found that in a slightly more
complicatedmodel,thecollapseoftheuniversewasverydifferentfrom
the expansion. I realized that I had made a mistake: the no boundary
conditionimpliedthatdisorderwouldinfactcontinuetoincreaseduring
the contraction. The thermodynamic and psychological arrows of time
would not reverse when the universe begins to recontract, or inside
blackholes.
What should you do when you find you have made a mistake like
that?Somepeopleneveradmitthattheyarewrongandcontinuetofind
new,andoftenmutuallyinconsistent,argumentstosupporttheircase—
as Eddington did in opposing black hole theory. Others claim to have
neverreallysupportedtheincorrectviewinthefirstplaceor,iftheydid,
itwasonlytoshowthatitwasinconsistent.Itseemstomemuchbetter
and less confusing if you admit in print that you were wrong. A good
example of this was Einstein, who called the cosmological constant,
whichheintroducedwhenhewastryingtomakeastaticmodelofthe
universe,thebiggestmistakeofhislife.
Toreturntothearrowoftime,thereremainsthequestion:whydowe
observe that the thermodynamic and cosmological arrows point in the
same direction? Or in other words, why does disorder increase in the
same direction of time as that in which the universe expands? If one
believesthattheuniversewillexpandandthencontractagain,astheno
boundaryproposalseemstoimply,thisbecomesaquestionofwhywe
shouldbeintheexpandingphaseratherthanthecontractingphase.
One can answer this on the basis of the weak anthropic principle.
Conditions in the contracting phase would not be suitable for the
existence of intelligent beings who could ask the question: why is
disorder increasing in the same direction of time as that in which the
universeisexpanding?Theinflationintheearlystagesoftheuniverse,
whichthenoboundaryproposalpredicts,meansthattheuniversemust
be expanding at very close to the critical rate at which it would just
avoidrecollapse,andsowillnotrecollapseforaverylongtime.Bythen
allthestarswillhaveburnedoutandtheprotonsandneutronsinthem
will probably have decayed into light particles and radiation. The
universewouldbeinastateofalmostcompletedisorder.Therewould
beno strong thermodynamicarrow oftime.Disorder couldn’tincrease
much because the universe would be in a state of almost complete
disorderalready.However,astrongthermodynamicarrowisnecessary
forintelligentlifetooperate.Inordertosurvive,humanbeingshaveto
consumefood,whichisanorderedformofenergy,andconvertitinto
heat, which is a disordered form of energy. Thus intelligent life could
notexistinthecontractingphaseoftheuniverse.Thisistheexplanation
ofwhyweobservethatthethermodynamicandcosmologicalarrowsof
time point in the same direction. It is not that the expansion of the
universecausesdisordertoincrease.Rather,itisthatthenoboundary
conditioncauses disorder toincreaseand the conditionstobe suitable
forintelligentlifeonlyintheexpandingphase.
To summarize, the laws of science do not distinguish between the
forward and backward directions of time. However, there are at least
threearrowsoftimethatdodistinguishthepastfromthefuture.They
are the thermodynamic arrow, the direction of time in which disorder
increases; the psychological arrow, the direction of time in which we
rememberthepastandnotthefuture;andthecosmologicalarrow,the
directionoftimeinwhichtheuniverseexpandsratherthancontracts.I
haveshownthatthepsychologicalarrowisessentiallythesameasthe
thermodynamicarrow,sothatthetwowouldalwayspointinthesame
direction. The no boundary proposal for the universe predicts the
existence of a well-defined thermodynamic arrow of time because the
universemuststartoffinasmoothandorderedstate.Andthereasonwe
observethisthermodynamicarrowtoagreewiththecosmologicalarrow
is that intelligent beings can exist only in the expanding phase. The
contracting phase will be unsuitable because it has no strong
thermodynamicarrowoftime.
The progress of the human race in understanding the universe has
established a small corner of order in an increasingly disordered
universe.Ifyouremembereveryword inthisbook, yourmemorywill
haverecordedabouttwomillionpiecesofinformation:theorderinyour
brain will have increased by about two million units. However, while
you have been reading the book, you will have converted at least a
thousandcaloriesoforderedenergy,intheformoffood,intodisordered
energy, in the form of heat that you lose to the air around you by
convectionandsweat.Thiswillincreasethedisorderoftheuniverseby
abouttwentymillionmillionmillionmillionunits—orabouttenmillion
millionmilliontimes theincreasein orderinyourbrain—andthat’sif
youremembereverythinginthisbook.InthenextchapterbutoneIwill
try to increase the order in our neck of the woods a little further by
explaining how people are trying to fit together the partial theories I
have described to form a complete unified theory that would cover
everythingintheuniverse.
T
CHAPTER10
WORMHOLESAND
TIMETRAVEL
helastchapterdiscussedwhyweseetimegoforward:whydisorder
increases and why we remember the past but not the future. Time
wastreatedasifitwereastraightrailwaylineonwhichonecouldonly
goonewayortheother.
But what if the railway line had loops and branches so that a train
could keep going forward but come back to a station it had already
passed?Inotherwords,mightitbepossibleforsomeonetotravelinto
thefutureorthepast?
H.G.Wellsin TheTimeMachine explored these possibilities as have
countless other writers of science fiction. Yet many of the ideas of
science fiction, like submarines and travel to the moon, have become
mattersofsciencefact.Sowhataretheprospectsfortimetravel?
Thefirstindicationthatthelawsofphysicsmightreallyallowpeople
totravelintimecamein1949whenKurtGödeldiscoveredanewspace-
timeallowedbygeneralrelativity.Gödelwasamathematicianwhowas
famous for proving that it is impossible to prove all true statements,
evenifyoulimityourselftotryingtoproveallthetruestatementsina
subject as apparently cut and dried as arithmetic. Like the uncertainty
principle, Gödel’s incompleteness theorem may be a fundamental
limitationonourabilitytounderstandandpredicttheuniverse,butso
far at least it hasn’t seemed to be an obstacle in our search for a
completeunifiedtheory.
GödelgottoknowaboutgeneralrelativitywhenheandEinsteinspent
their later years at the Institute for Advanced Study in Princeton. His
space-time had the curious property that the whole universe was
rotating.Onemightask:“Rotatingwithrespecttowhat?”Theansweris
thatdistantmatterwouldberotatingwithrespecttodirectionsthatlittle
topsorgyroscopespointin.
Thishadthesideeffectthatitwouldbepossibleforsomeonetogooff
in a rocket ship and return to earth before he set out. This property
reallyupsetEinstein, whohadthought thatgeneral relativitywouldn’t
allow time travel. However, given Einstein’s record of ill-founded
oppositiontogravitationalcollapseandtheuncertaintyprinciple,maybe
this was an encouraging sign. The solution Gödel found doesn’t
correspond to the universe we live in because we can show that the
universeisnotrotating.Italsohadanon-zerovalueofthecosmological
constant that Einstein introduced when he thought the universe was
unchanging. After Hubble discovered the expansion of the universe,
therewasnoneedfor acosmological constantand itis nowgenerally
believed to be zero. However, other more reasonable space-times that
areallowedbygeneralrelativityandwhichpermittravelintothepast
have since been found. One is in the interior of a rotating black hole.
Another is a space-time that contains two cosmic strings moving past
each other at high speed. As their name suggests, cosmic strings are
objects that are like string in that they have length but a tiny cross
section. Actually, they are more like rubber bands because they are
underenormoustension,somethinglikeamillionmillionmillionmillion
tons.Acosmicstringattachedtotheearthcouldaccelerateitfrom0to
60 mph in 1/30th of a second. Cosmic strings may sound like pure
sciencefictionbuttherearereasonstobelievetheycouldhaveformed
in the early universe as a result of symmetry-breaking of the kind
discussedinChapter5.Becausetheywouldbeunderenormoustension
andcouldstartinanyconfiguration,theymightacceleratetoveryhigh
speedswhentheystraightenout.
The Gödel solution and the cosmic string space-time start out so
distortedthattravelintothepastwasalwayspossible.Godmighthave
createdsuchawarpeduniversebutwehavenoreasontobelievehedid.
Observationsofthemicrowavebackgroundandoftheabundancesofthe
lightelementsindicatethattheearlyuniversedidnothavethekindof
curvaturerequiredtoallowtimetravel.Thesameconclusionfollowson
theoretical grounds if the no boundary proposal is correct. So the
question is: if the universe starts out without the kind of curvature
required for time travel, can we subsequently warp local regions of
space-timesufficientlytoallowit?
Acloselyrelatedproblemthatisalsoofconcerntowritersofscience
fictionisrapidinterstellarorintergalactictravel.Accordingtorelativity,
nothingcantravelfasterthanlight.Ifwethereforesentaspaceshipto
ournearestneighboringstar,AlphaCentauri,whichisaboutfourlight-
yearsaway,itwouldtakeatleasteightyearsbeforewecouldexpectthe
travelers to return and tell us what they had found. If the expedition
weretothecenterofourgalaxy,itwouldbeatleastahundredthousand
years before it came back. The theory of relativity does allow one
consolation.Thisistheso-calledtwinsparadoxmentionedinChapter2.
Becausethereisnouniquestandardoftime,butratherobserverseach
havetheirowntimeasmeasuredbyclocksthattheycarrywiththem,it
is possible for the journey to seem to be much shorter for the space
travelers than for those who remain on earth. But there would not be
muchjoyinreturningfromaspacevoyageafewyearsoldertofindthat
everyoneyouhadleftbehindwasdeadandgonethousandsofyearsago.
Soin orderto haveany humaninterest intheir stories,science fiction
writers had to suppose that we would one day discover how to travel
fasterthanlight.Whatmostoftheseauthorsdon’tseemtohaverealized
isthatifyoucantravelfasterthanlight,thetheoryofrelativityimplies
youcanalsotravelbackintime,asthefollowinglimericksays:
TherewasayoungladyofWight
Whotravelledmuchfasterthanlight.
Shedepartedoneday,
Inarelativeway,
Andarrivedonthepreviousnight.
Thepointisthatthetheoryofrelativitysaysthatthereisnounique
measureoftimethatallobserverswillagreeon.Rather,eachobserver
hashisorherownmeasureoftime.Ifitispossibleforarockettraveling
belowthespeedoflighttogetfromeventA(say,thefinalofthe100-
meterraceoftheOlympicGamesin2012)toeventB(say,theopening
of the 100,004th meeting of the Congress of Alpha Centauri), then all
observerswillagreethateventAhappenedbeforeeventBaccordingto
theirtimes.Suppose,however,thatthespaceshipwouldhavetotravel
faster than light to carry the news of the race to the Congress. Then
observersmovingatdifferentspeedscandisagreeaboutwhethereventA
occurred before B or vice versa. According to the time of an observer
who is at rest with respect to the earth, it may be that the Congress
openedaftertherace.Thusthisobserverwouldthinkthataspaceship
couldgetfromAtoBintimeifonlyitcouldignorethespeed-of-light
speed limit. However, to an observer at Alpha Centauri moving away
fromtheearthatnearlythespeedoflight,itwouldappearthateventB,
theopeningoftheCongress,wouldoccurbeforeeventA,the100-meter
race. The theory of relativity says that the laws of physics appear the
sametoobserversmovingatdifferentspeeds.
This has been well tested by experiment and is likely to remain a
feature even if we find a more advanced theory to replace relativity.
Thus the moving observer would say that if faster-than-light travel is
possible, it should be possible to get from event B, theopening of the
Congress,toeventA,the100-meterrace.Ifonewentslightlyfaster,one
could even get back before the race and place a bet on it in the sure
knowledgethatonewouldwin.
There is a problem with breaking the speed-of-light barrier. The
theory of relativity says that the rocket power needed to accelerate a
spaceshipgetsgreaterandgreatertheneareritgetstothespeedoflight.
We have experimental evidence for this, not with spaceships but with
elementary particles in particle accelerators like those at Fermilab or
CERN (European Centre for Nuclear Research). We can accelerate
particlesto99.99percentofthespeedoflight,buthowevermuchpower
we feed in, we can’t get them beyond the speed-of-light barrier.
Similarlywithspaceships:nomatterhowmuchrocketpowertheyhave,
theycan’tacceleratebeyondthespeedoflight.
Thatmightseemtoruleoutbothrapidspacetravelandtravelbackin
time.However,thereisapossiblewayout.Itmightbethatonecould
warpspace-timesothattherewasashortcutbetweenAandB.Oneway
of doing this would be to create a wormhole between A and B. As its
name suggests, a wormhole is a thin tube of space-time which can
connecttwonearlyflatregionsfarapart.
Thereneedbenorelationbetweenthedistancethroughthewormhole
andthe separationofits ends inthe nearly flatbackground.Thus one
couldimaginethatonecouldcreateorfindawormholethatwouldlead
from the vicinity of the Solar System to Alpha Centauri. The distance
throughthe wormhole might be only a few millionmileseven though
earth and Alpha Centauri are twenty million million miles apart in
ordinaryspace. Thiswould allownews of the100-meter racetoreach
the opening of the Congress. But then an observer moving toward the
earthshouldalso beableto findanotherwormholethatwould enable
himtogetfromtheopeningoftheCongressonAlphaCentauribackto
earthbeforethestartoftherace.Sowormholes,likeanyotherpossible
formoftravelfasterthanlight,wouldallowonetotravelintothepast.
The idea of wormholes between different regions of space-time was
not an invention of science fiction writers but came from a very
respectablesource.
In 1935, Einstein and Nathan Rosen wrote a paper in which they
showed that general relativity allowed what they called “bridges,” but
whicharenowknownaswormholes.TheEinstein-Rosenbridgesdidn’t
lastlongenoughforaspaceshiptogetthrough:theshipwouldruninto
a singularity as the wormhole pinched off. However, it has been
suggestedthatitmightbepossibleforanadvancedcivilizationtokeepa
wormholeopen.Todothis,ortowarpspace-timeinanyotherwaysoas
to permit time travel, one can show that one needs aregion of space-
time with negative curvature, like the surface of a saddle. Ordinary
matter,whichhasapositiveenergydensity,givesspace-timeapositive
curvature,likethesurfaceofasphere.So whatoneneeds,inorderto
warpspace-timeinawaythatwillallowtravelintothepast,ismatter
withnegativeenergydensity.
Energy is a bit like money: if you have a positive balance, you can
distribute it in various ways, but according to the classical laws that
werebelievedatthebeginningofthecentury,youweren’tallowedtobe
overdrawn.Sotheseclassicallawswouldhaveruledoutanypossibility
oftime travel.However, ashasbeen describedin earlier chapters,the
classical laws were superseded by quantum laws based on the
uncertaintyprinciple.Thequantumlawsaremoreliberalandallowyou
tobeoverdrawn on one or two accounts provided the total balanceis
positive.Inotherwords,quantumtheoryallowstheenergydensitytobe
negative in some places, provided that this is made up for by positive
energy densities in other places, so that the total energy remains
positive.Anexampleofhowquantumtheorycanallownegativeenergy
densitiesisprovidedbywhatiscalledtheCasimireffect.Aswesawin
Chapter7,evenwhatwethinkofas“empty”spaceisfilledwithpairsof
virtualparticlesandantiparticlesthatappeartogether,moveapart,and
come back together and annihilate each other. Now, suppose one has
twoparallelmetalplatesashortdistanceapart.Theplateswillactlike
mirrorsforthevirtualphotonsorparticlesoflight.Infacttheywillform
acavitybetweenthem,abitlikeanorganpipethatwillresonateonlyat
certain notes. This means that virtual photons can occur in the space
betweentheplatesonlyiftheirwavelengths(thedistancebetweenthe
crestofonewaveandthenext)fitawholenumberoftimesintothegap
between the plates. If the width of a cavity is a whole number of
wavelengthsplusafractionofawavelength,thenaftersomereflections
backwardand forwardbetween theplates, the crestsof onewavewill
coincidewiththetroughsofanotherandthewaveswillcancelout.
Because the virtual photons between the plates can have only the
resonant wavelengths, there will be slightly fewer of them than in the
region outside the plates where virtual photons can have any
wavelength.Thustherewillbeslightlyfewervirtualphotonshittingthe
inside surfaces of the plates than the outside surfaces. One would
thereforeexpectaforceontheplates,pushingthemtowardeachother.
Thisforcehasactuallybeendetectedandhasthepredictedvalue.Thus
wehaveexperimentalevidencethatvirtualparticlesexistandhavereal
effects.
Thefactthattherearefewervirtualphotonsbetweentheplatesmeans
thattheirenergydensitywillbelessthanelsewhere.Butthetotalenergy
densityin“empty”spacefarawayfromtheplatesmustbezero,because
otherwisetheenergydensitywouldwarpthespaceanditwouldnotbe
almostflat.So,iftheenergydensitybetweentheplatesislessthanthe
energydensityfaraway,itmustbenegative.
We thus have experimental evidence both that space-time can be
warped (from the bending of light during eclipses) and that it can be
curved in the way necessary to allow time travel (from the Casimir
effect). One might hope therefore that as we advance in science and
technology,wewouldeventuallymanagetobuildatimemachine.Butif
so,whyhasn’tanyonecomebackfromthefutureandtoldushowtodo
it?Theremightbegoodreasonswhyitwouldbeunwisetogiveusthe
secretoftimetravelatourpresentprimitivestateofdevelopment,but
unlesshumannaturechangesradically,itisdifficulttobelievethatsome
visitorfromthefuturewouldn’tspillthebeans.Ofcourse,somepeople
would claim that sightings of UFOs are evidence that we are being
visitedeitherbyaliensorbypeoplefromthefuture.(Ifthealienswere
togethereinreasonabletime,theywouldneedfaster-than-lighttravel,
sothetwopossibilitiesmaybeequivalent.)
However, I think that any visit by aliens or people from the future
wouldbemuchmoreobviousand,probably,muchmoreunpleasant.If
theyaregoingtorevealthemselvesatall,whydosoonlytothosewho
arenotregardedasreliablewitnesses?Iftheyaretryingtowarnusof
somegreatdanger,theyarenotbeingveryeffective.
A possible way to explain the absence of visitors from the future
wouldbetosaythatthepastisfixedbecausewehaveobserveditand
seen that it does not have the kind of warping needed to allow travel
back from the future. On the other hand, the future is unknown and
open, so it might well have the curvature required. This would mean
thatanytimetravelwouldbeconfinedtothefuture.Therewouldbeno
chance of Captain Kirk and the Starship Enterprise turning up at the
presenttime.
Thismightexplainwhywehavenotyetbeenoverrunbytouristsfrom
thefuture,butitwouldnotavoidtheproblemsthatwouldariseifone
were able to go back and change history. Suppose, for example, you
went back and killed your great-great-grandfather while he was still a
child.Therearemanyversionsofthisparadoxbuttheyareessentially
equivalent:onewouldgetcontradictionsifonewerefreetochangethe
past.
Thereseemtobetwopossibleresolutionstotheparadoxesposedby
timetravel.OneIshallcalltheconsistenthistoriesapproach.Itsaysthat
evenifspace-timeiswarpedsothatitwouldbepossibletotravelinto
thepast,whathappensinspace-timemustbeaconsistentsolutionofthe
lawsofphysics.Accordingtothisviewpoint,youcouldnotgobackin
timeunlesshistoryshowedthatyouhadalreadyarrivedinthepastand,
while there, had not killed your great-great-grandfather or committed
any other acts that would conflict with your current situation in the
present. Moreover, when you did go back, you wouldn’t be able to
changerecordedhistory.Thatmeansyouwouldn’thavefreewilltodo
whatyouwanted.Ofcourse,onecouldsaythatfreewillisanillusion
anyway. If there really is a complete unified theory that governs
everything,itpresumablyalsodeterminesyouractions.Butitdoessoin
a way that is impossible to calculate for an organism that is as
complicatedasahumanbeing.Thereasonwesaythathumanshavefree
will is because we can’t predict what they will do. However, if the
humanthengoesoffinarocketshipandcomesbackbeforeheorsheset
off,wewillbeabletopredictwhatheorshewilldobecauseitwillbe
partofrecordedhistory.Thus,inthatsituation,thetimetravelerwould
havenofreewill.
Theotherpossiblewaytoresolvetheparadoxesoftimetravelmight
becalledthealternativehistorieshypothesis.Theideahereisthatwhen
timetravelersgobacktothepast,theyenteralternativehistorieswhich
differ from recorded history. Thus they can act freely, without the
constraint of consistency with their previous history. Steven Spielberg
hadfunwiththisnotionintheBacktotheFuturefilms:MartyMcFlywas
abletogobackandchangehisparents’courtshiptoamoresatisfactory
history.
The alternative histories hypothesis sounds rather like Richard
Feynman’s way of expressing quantum theory as a sum over histories,
which was described in Chapters 4 and 8. This said that the universe
didn’t just have a single history: rather it had every possible history,
eachwithitsownprobability.However,thereseemstobeanimportant
difference between Feynman’s proposal and alternative histories. In
Feynman’s sum, each history comprises a complete space-time and
everythinginit.Thespace-timemaybesowarpedthatitispossibleto
travelinarocketintothepast.Buttherocketwouldremaininthesame
space-time and therefore the same history, which would have to be
consistent.ThusFeynman’ssumoverhistoriesproposalseemstosupport
theconsistenthistorieshypothesisratherthanthealternativehistories.
TheFeynmansumoverhistoriesdoesallowtravelintothepastona
microscopic scale. In Chapter 9 we saw that the laws of science are
unchangedby combinations of the operationsC,P, and T. This means
thatanantiparticlespinningintheanticlockwisedirectionandmoving
from A to B can also be viewed as an ordinary particle spinning
clockwise and moving backward in time from B to A. Similarly, an
ordinaryparticlemovingforwardintimeisequivalenttoanantiparticle
moving backward in time. As has been discussed in this chapter and
Chapter 7, “empty” space is filled with pairs of virtual particles and
antiparticles that appear together, move apart, and then come back
togetherandannihilateeachother.
So,onecanregardthepairofparticlesasasingleparticlemovingon
a closed loop in space-time. When the pair is moving forward in time
(fromtheeventatwhichitappearstothatatwhichitannihilates),itis
calledaparticle.Butwhentheparticleistravelingbackintime(from
theeventatwhichthepairannihilatestothatatwhichitappears),itis
saidtobeanantiparticletravelingforwardintime.
Theexplanationof howblackholes canemit particlesand radiation
(given in Chapter 7) was that one member of a virtual
particle/antiparticlepair(say,theantiparticle)mightfallintotheblack
hole, leaving the other member without a partner with which to
annihilate.Theforsakenparticlemightfallintotheholeaswell,butit
might also escape from the vicinity of the black hole. If so, to an
observer at a distance it would appear to be aparticle emitted by the
blackhole.
Onecan,however,haveadifferentbutequivalentintuitivepictureof
the mechanism for emission from black holes. One can regard the
member of the virtual pair that fell into the black hole (say, the
antiparticle) as a particle traveling backward in time out of the hole.
Whenitgetstothepointatwhichthevirtualparticle/antiparticlepair
appearedtogether,itisscatteredbythegravitationalfieldintoaparticle
travelingforwardintimeandescapingfromtheblackhole.If,instead,it
weretheparticlememberofthevirtualpairthatfellintothehole,one
couldregarditasanantiparticletravelingbackintimeandcomingout
oftheblackhole.Thustheradiationbyblackholesshowsthatquantum
theoryallowstravelbackintimeonamicroscopicscaleandthatsuch
timetravelcanproduceobservableeffects.
One can therefore ask: does quantum theory allow time travel on a
macroscopic scale, which people could use? At first sight, it seems it
should.TheFeynmansumoverhistoriesproposalissupposedtobeover
all histories. Thus it should include histories in which space-time is so
warpedthatitispossibletotravelintothepast.Whythenaren’twein
troublewithhistory?Suppose,forexample,someonehadgonebackand
giventheNazisthesecretoftheatombomb?
One would avoid these problems if what I call the chronology
protectionconjectureholds.Thissaysthatthelawsofphysicsconspire
to prevent macroscopic bodies from carrying information into the past.
Likethecosmiccensorshipconjecture,ithasnotbeenprovedbutthere
arereasonstobelieveitistrue.
Thereasontobelievethatchronologyprotectionoperatesisthatwhen
space-time is warped enough to make travel into the past possible,
virtualparticlesmovingonclosedloopsinspace-timecanbecomereal
particles traveling forward in time at or below the speed of light. As
these particles can go round the loop any number of times, they pass
eachpointontheirroutemanytimes.Thustheirenergyiscountedover
and over again and the energy density will become very large. This
couldgive space-time a positivecurvature that would notallow travel
into the past. It is not yet clear whether these particles would cause
positive or negative curvature or whether the curvature produced by
some kinds of virtual particles might cancel that produced by other
kinds.Thusthepossibilityoftimetravelremainsopen.ButI’mnotgoing
tobetonit.Myopponentmighthavetheunfairadvantageofknowing
thefuture.
A
CHAPTER11
THEUNIFICATION
OFPHYSICS
s was explained in the first chapter, it would be very difficult to
constructacompleteunifiedtheoryofeverythingintheuniverseall
atonego.Soinsteadwehavemadeprogressbyfindingpartialtheories
that describe a limited range of happenings and by neglecting other
effects or approximating them by certain numbers. (Chemistry, for
example, allows us to calculate the interactions of atoms, without
knowing the internal structure of an atom’s nucleus.) Ultimately,
however,onewouldhopetofindacomplete,consistent,unifiedtheory
thatwouldincludeallthesepartialtheoriesasapproximations,andthat
did not need to be adjusted to fit the facts by picking the values of
certainarbitrarynumbersinthetheory.Thequestforsuchatheoryis
known as “the unification of physics.” Einstein spent most of his later
yearsunsuccessfullysearchingforaunifiedtheory,butthetimewasnot
ripe: there were partial theories for gravity and the electromagnetic
force, but very little was known about the nuclear forces. Moreover,
Einsteinrefusedtobelieveintherealityofquantummechanics,despite
theimportantrolehehadplayedinitsdevelopment.Yetitseemsthat
theuncertaintyprincipleisafundamentalfeatureoftheuniversewelive
in. A successful unified theory must, therefore, necessarily incorporate
thisprinciple.
AsIshalldescribe,theprospectsforfindingsuchatheoryseemtobe
muchbetter now because we know so muchmoreabout the universe.
But we must beware of overconfidence—we have had false dawns
before!Atthebeginningofthiscentury,forexample,itwasthoughtthat
everythingcouldbeexplainedintermsofthepropertiesofcontinuous
matter,suchaselasticityandheatconduction.Thediscoveryofatomic
structure and the uncertainty principle put an emphatic end to that.
Thenagain,in1928,physicistandNobelPrizewinnerMaxBorntolda
groupofvisitorstoGöttingenUniversity,“Physics,asweknowit,willbe
overinsixmonths.”Hisconfidencewasbasedontherecentdiscovery
byDiracoftheequationthatgovernedtheelectron.Itwasthoughtthat
asimilarequationwouldgoverntheproton,whichwastheonlyother
particle known at the time, and that would be the end of theoretical
physics. However, the discovery of the neutron and of nuclear forces
knockedthatoneontheheadtoo.Havingsaidthis,Istillbelievethere
aregroundsforcautiousoptimismthatwemaynowbeneartheendof
thesearchfortheultimatelawsofnature.
In previous chapters I have described general relativity, the partial
theory of gravity, and the partial theories that govern the weak, the
strong,andtheelectromagneticforces.Thelastthreemaybecombined
in so-called grand unified theories, or GUTs, which are not very
satisfactory because they do not include gravity and because they
contain a number of quantities, like the relative masses of different
particles,thatcannotbepredictedfromthetheorybuthavetobechosen
to fit observations. The main difficulty in finding a theory that unifies
gravity with the other forces is that general relativity is a “classical”
theory; that is, it does not incorporate the uncertainty principle of
quantum mechanics. On the other hand, the other partial theories
depend on quantum mechanics in an essential way. A necessary first
step, therefore, is to combine general relativity with the uncertainty
principle. As we have seen, this can produce some remarkable
consequences,suchasblackholesnotbeingblack,andtheuniversenot
havinganysingularitiesbutbeingcompletelyself-containedandwithout
a boundary. The trouble is, as explained in Chapter 7, that the
uncertaintyprinciplemeansthateven“empty”spaceisfilledwithpairs
ofvirtualparticlesandantiparticles.Thesepairswouldhaveaninfinite
amountofenergyand,therefore,byEinstein’sfamousequationE=mc2,
they would have an infinite amount of mass. Their gravitational
attractionwouldthuscurveuptheuniversetoinfinitelysmallsize.
Rather similar, seemingly absurd infinities occur in the other partial
theories, but in all these cases the infinities can be canceled out by a
processcalledrenormalization.Thisinvolvescancelingtheinfinitiesby
introducing other infinities. Although this technique is rather dubious
mathematically, it does seem to work in practice, and has been used
withthesetheoriestomakepredictionsthatagreewithobservationsto
an extraordinary degree of accuracy. Renormalization, however, does
have a serious drawback from the point of view of trying to find a
completetheory,becauseitmeansthattheactualvaluesofthemasses
andthestrengthsoftheforcescannotbepredictedfromthetheory,but
havetobechosentofittheobservations.
In attempting to incorporate the uncertainty principle into general
relativity,onehasonlytwoquantitiesthatcanbeadjusted:thestrength
of gravity and the value of the cosmological constant. But adjusting
these is not sufficient to remove all the infinities. One therefore has a
theory that seems to predict that certain quantities, such as the
curvature of space-time, are really infinite, yet these quantities can be
observedandmeasuredtobeperfectlyfinite!Thisproblemincombining
generalrelativity andthe uncertainty principlehad been suspectedfor
sometime,butwasfinallyconfirmedbydetailedcalculationsin1972.
Four years later, a possible solution, called “supergravity,” was
suggested. The idea was to combine the spin-2 particle called the
graviton, which carries the gravitational force, with certain other
particles of spin 3/2, 1, ½, and 0. In a sense, all these particles could
then be regarded as different aspects of the same “superparticle,” thus
unifyingthematterparticleswithspin½and3/2withtheforce-carrying
particlesofspin0,1,and2.Thevirtualparticle/antiparticlepairsofspin
½and3/2wouldhavenegativeenergy,andsowouldtendtocancelout
thepositiveenergyofthespin2,1,and0virtualpairs.Thiswouldcause
manyofthe possibleinfinities tocancel out,but itwas suspectedthat
someinfinitiesmightstillremain.However,thecalculationsrequiredto
findoutwhetherornottherewereanyinfinitiesleftuncanceledwereso
long and difficult that no one was prepared to undertake them. Even
withacomputeritwasreckoneditwouldtakeatleastfouryears,and
thechanceswereveryhighthatonewouldmakeatleastonemistake,
probably more. So one would know one had the right answer only if
someoneelserepeatedthecalculationandgotthesameanswer,andthat
didnotseemverylikely!
Despite these problems, and the fact that the particles in the
supergravitytheoriesdidnotseemtomatchtheobservedparticles,most
scientists believed that supergravity was probably the right answer to
the problem of the unification of physics. It seemed the best way of
unifying gravity with the other forces. However, in 1984 there was a
remarkablechangeofopinioninfavorofwhatarecalledstringtheories.
In these theories the basic objects are not particles, which occupy a
single point of space, but things that have a length but no other
dimension,likeaninfinitelythinpieceofstring.Thesestringsmayhave
ends (the so-called open strings) or they may be joined up with
themselvesinclosedloops(closed strings)(Fig.11.1 and Fig. 11.2). A
particle occupies one point of space at each instant of time. Thus its
historycanberepresentedbyalineinspace-time(the“world-line”).A
string, on the other hand, occupies a line in space at each moment of
time.Soitshistoryinspace-timeisatwo-dimensionalsurfacecalledthe
world-sheet.(Anypointonsuchaworld-sheetcanbedescribedbytwo
numbers,onespecifyingthetimeandtheotherthepositionofthepoint
on the string.) The world-sheet of an open string is a strip: its edges
represent the paths through space-time of the ends of the string (Fig.
11.1).Theworld-sheetofaclosedstringisacylinderortube(Fig.11.2):
aslicethroughthetubeisacircle,whichrepresentsthepositionofthe
stringatoneparticulartime.
Twopiecesofstringcanjointogethertoformasinglestring;inthe
caseofopenstringstheysimplyjoinattheends(Fig.11.3),whileinthe
caseofclosedstringsitislikethetwolegsjoiningonapairoftrousers
(Fig.11.4).Similarly,asinglepieceofstringcandivideintotwostrings.
Instringtheories,whatwerepreviouslythoughtofasparticlesarenow
picturedaswavestravelingdownthestring,likewavesonavibrating
kite string. The emission or absorption of one particle by another
correspondstothedividingorjoiningtogetherofstrings.Forexample,
thegravitationalforceofthesunontheearthwaspicturedinparticle
theoriesasbeingcausedbytheemissionofagravitonbyaparticlein
thesunanditsabsorptionbyaparticleintheearth(Fig.11.5).Instring
theory,thisprocesscorrespondstoanH-shapedtubeorpipe(Fig.11.6)
(stringtheoryisratherlikeplumbing,inaway).Thetwoverticalsides
of the H correspond to the particles in the sun and the earth, and the
horizontal crossbar corresponds to the graviton that travels between
them.
FIGURE11.1ANDFIGURE11.2
FIGURE11.3
Stringtheoryhasacurioushistory. Itwasoriginally inventedinthe
late 1960s in an attempt to find a theory to describe the strong force.
The idea was that particles like the proton and the neutron could be
regardedaswaveson astring.Thestrongforces betweentheparticles
would correspond to pieces of string that went between other bits of
string,asinaspidersweb.Forthistheorytogivetheobservedvalueof
the strong force between particles, the strings had to be like rubber
bandswithapullofabouttentons.
FIGURE11.4
In1974JoëlScherkfromParisandJohnSchwarzfromtheCalifornia
Institute of Technology published a paper in which they showed that
string theory could describe the gravitational force, but only if the
tensioninthestringwereverymuchhigher,aboutathousandmillion
million million million million million tons (1 with thirty-nine zeros
afterit).Thepredictionsofthestringtheorywouldbejustthesameas
thoseofgeneralrelativityonnormallengthscales,buttheywoulddiffer
at very small distances, less than a thousand million million million
millionmillionthofacentimeter(acentimeterdividedby1withthirty-
three zeros after it). Their work did not receive much attention,
however, because at just about that time most people abandoned the
originalstringtheoryofthestrongforceinfavorofthetheorybasedon
quarksandgluons,whichseemedtofitmuchbetterwithobservations.
Scherkdiedintragiccircumstances(hesufferedfromdiabetesandwent
into a coma when no one was around to give him an injection of
insulin).SoSchwarzwasleftaloneasalmosttheonlysupporterofstring
theory, but now with the much higher proposed value of the string
tension.
FIGURE11.5ANDFIGURE11.6
In 1984 interest in strings suddenly revived, apparently for two
reasons. One was that people were not really making much progress
towardshowingthatsupergravitywasfiniteorthatitcouldexplainthe
kinds of particles that we observe. The other was the publication of a
paperbyJohnSchwarzandMikeGreenofQueenMaryCollege,London,
thatshowedthatstringtheorymightbeabletoexplaintheexistenceof
particlesthathaveabuilt-inleft-handedness,likesomeoftheparticles
thatweobserve.Whateverthereasons,alargenumberofpeoplesoon
begantoworkonstringtheoryandanewversionwasdeveloped,theso-
calledheteroticstring,whichseemedasifitmightbeabletoexplainthe
typesofparticlesthatweobserve.
String theories also lead to infinities, but it is thought they will all
cancel out in versions like the heterotic string (though this is not yet
known for certain). String theories, however, have a bigger problem:
theyseemtobeconsistentonlyifspace-timehaseithertenortwenty-six
dimensions, instead of the usual four! Of course, extra space-time
dimensionsareacommonplaceofsciencefictionindeed,theyprovidean
idealwayofovercomingthenormalrestrictionofgeneralrelativitythat
onecannottravelfasterthanlightorbackintime(seeChapter10).The
ideaistotakeashortcutthroughtheextradimensions.Onecanpicture
thisinthefollowingway.Imaginethatthespaceweliveinhasonlytwo
dimensionsandiscurvedlikethesurfaceofananchorringortorus(Fig.
11.7). If you were on one side of the inside edge of the ring and you
wantedtogettoapointontheotherside,youwouldhavetogoround
the inner edge of the ring. However, if you were able to travel in the
thirddimension,youcouldcutstraightacross.
FIGURE11.7
Why don’t we notice all these extra dimensions, if they are really
there? Why do we see only three space dimensions and one time
dimension? The suggestion is that the other dimensions are curved up
intoaspaceofverysmallsize,somethinglikeamillionmillionmillion
millionmillionthofaninch.Thisissosmallthatwejustdon’tnoticeit:
weseeonlyonetimedimensionandthreespacedimensions,inwhich
space-timeisfairlyflat.Itislikethesurfaceofastraw.Ifyoulookatit
closely, you see it is two-dimensional (the position of a point on the
strawisdescribedbytwonumbers,thelengthalongthestrawandthe
distance round the circular direction). But if you look at it from a
distance, you don’t see the thickness of the straw and it looks one-
dimensional(thepositionofapointisspecifiedonlybythelengthalong
the straw). So it is with space-time: on a very small scale it is ten-
dimensional and highly curved, but on bigger scales you don’t see the
curvatureortheextradimensions.Ifthispictureiscorrect,itspellsbad
newsforwould-bespacetravelers:theextradimensionswouldbefartoo
small to allow a spaceship through. However, it raises another major
problem.Whyshouldsome,butnotall,ofthedimensionsbecurledup
into a small ball? Presumably, in the very early universe all the
dimensionswouldhavebeenverycurved.Whydidonetimedimension
and three space dimensions flatten out, while the other dimensions
remaintightlycurledup?
Onepossibleansweristheanthropicprinciple.Twospacedimensions
donotseemtobeenoughtoallowforthedevelopmentofcomplicated
beings like us. For example, two-dimensional animals living on a one-
dimensionalearthwouldhavetoclimbovereachotherinordertoget
pasteachother.Ifatwo-dimensionalcreatureatesomethingitcouldnot
digestcompletely,itwouldhavetobringuptheremainsthesamewayit
swallowedthem,becauseiftherewereapassagerightthroughitsbody,
it would divide the creature into two separate halves: our two-
dimensionalbeingwouldfallapart(Fig.11.8).Similarly,itisdifficultto
seehowtherecouldbeanycirculationofthebloodinatwo-dimensional
creature.
Therewouldalsobeproblemswithmorethanthreespacedimensions.
Thegravitationalforcebetweentwobodieswoulddecreasemorerapidly
withdistancethanitdoesinthreedimensions.(Inthreedimensions,the
gravitational force drops to ¼ if one doubles the distance. In four
dimensionsitwoulddropto⅛,infivedimensionsto1/16,andsoon.)
The significance of this is that the orbits of planets, like the earth,
aroundthesunwouldbeunstable:theleastdisturbancefromacircular
orbit(suchaswouldbecausedbythegravitationalattractionofother
planets)wouldresultintheearthspiralingawayfromorintothesun.
Wewouldeitherfreezeorbeburnedup.Infact,thesamebehaviorof
gravity with distance in more than three space dimensions means that
the sun would not be able to exist in a stable state with pressure
balancinggravity.Itwouldeitherfallapartoritwouldcollapsetoform
ablackhole.Ineithercase,itwouldnotbeofmuchuseasasourceof
heatandlightforlifeonearth.Onasmallerscale,theelectricalforces
that cause the electrons to orbit round the nucleus in an atom would
behaveinthesamewayasgravitationalforces.Thustheelectronswould
eitherescapefromtheatomaltogetherorwouldspiralintothenucleus.
Ineithercase,onecouldnothaveatomsasweknowthem.
FIGURE11.8
Itseemsclearthenthatlife,atleastasweknowit,canexistonlyin
regions of space-time in which one time dimension and three space
dimensions are not curled up small. This would mean that one could
appeal to the weak anthropic principle, provided one could show that
stringtheorydoesatleastallowtheretobesuchregionsoftheuniverse
—anditseemsthatindeedstringtheorydoes.Theremaywellbeother
regionsoftheuniverse,orotheruniverses(whateverthatmaymean),in
whichallthedimensionsarecurledupsmallorinwhichmorethanfour
dimensionsare nearlyflat, buttherewould beno intelligentbeingsin
suchregionstoobservethedifferentnumberofeffectivedimensions.
Anotherproblemisthatthereareatleastfourdifferentstringtheories
(openstringsandthreedifferentclosedstringtheories)andmillionsof
waysinwhichtheextradimensionspredictedbystringtheorycouldbe
curledup.Whyshouldjustonestringtheoryandonekindofcurlingup
be picked out? For a time there seemed no answer, and progress got
boggeddown.Then,fromabout1994,peoplestarteddiscoveringwhat
arecalleddualities:differentstringtheoriesanddifferentwaysofcurling
up the extra dimensions could lead to the same results in four
dimensions.Moreover,aswellasparticles,whichoccupyasinglepoint
of space, and strings, which are lines, there were found to be other
objects called p-branes, which occupied two-dimensional or higher-
dimensionalvolumesinspace.(Aparticlecanberegardedasa0-brane
andastringasa1-branebuttherewerealsop-branesforp=2top=9.)
Whatthisseemstoindicateisthatthereisasortofdemocracyamong
supergravity,string,andp-branetheories:theyseemtofittogetherbut
nonecanbesaidtobemorefundamentalthantheothers.Theyappear
to be different approximations to some fundamental theory that are
validindifferentsituations.
People have searched for this underlying theory, but without any
success so far. However, I believe there may not be any single
formulationofthefundamentaltheoryanymorethan,asGödelshowed,
onecouldformulatearithmeticintermsofasinglesetofaxioms.Instead
itmaybelikemaps—youcan’tuseasinglemaptodescribethesurface
oftheearthorananchorring:youneedatleasttwomapsinthecaseof
theearthandfourfortheanchorringtocovereverypoint.Eachmapis
validonlyinalimitedregion,butdifferentmapswillhavearegionof
overlap.Thecollection ofmapsprovidesa completedescription ofthe
surface. Similarly, in physics it may be necessary to use different
formulations in different situations, but two different formulations
would agree in situations where they can both be applied. The whole
collection of different formulations could be regarded as a complete
unified theory, though one that could not be expressed in terms of a
singlesetofpostulates.
Butcantherereallybesuchaunifiedtheory?Orareweperhapsjust
chasingamirage?Thereseemtobethreepossibilities:
1. There really is a complete unified theory (or a collection of
overlapping formulations), which we will someday discover if we
aresmartenough.
2. Thereisnoultimatetheoryoftheuniverse,justaninfinitesequence
oftheoriesthatdescribetheuniversemoreandmoreaccurately.
3. There is no theory of the universe: events cannot be predicted
beyond a certain extent but occur in a random and arbitrary
manner.
Somewouldargueforthethirdpossibilityonthegroundsthatifthere
were a complete set of laws, that would infringe God’s freedom to
change his mind and intervene in the world. It’s a bit like the old
paradox: can God make a stone so heavy that he can’t lift it? But the
idea that God might want to change his mind is an example of the
fallacy, pointed out by St. Augustine, of imagining God as a being
existingintime:timeisapropertyonlyoftheuniversethatGodcreated.
Presumably,heknewwhatheintendedwhenhesetitup!
With the advent of quantum mechanics, we have come to recognize
thateventscannotbepredictedwithcompleteaccuracybutthatthereis
always a degree of uncertainty. If one likes, one could ascribe this
randomnesstotheinterventionofGod,butitwouldbeaverystrange
kindofintervention:thereisnoevidencethatitisdirectedtowardany
purpose. Indeed, if it were, it would by definition not be random. In
modern times, we have effectively removed the third possibility above
byredefiningthegoalofscience:ouraimistoformulateasetoflaws
that enables us to predict events only up to the limit set by the
uncertaintyprinciple.
Thesecondpossibility,thatthereisaninfinitesequenceofmoreand
morerefinedtheories,isinagreementwithallourexperiencesofar.On
manyoccasionswehaveincreasedthesensitivityofourmeasurements
ormadeanewclassofobservations,onlytodiscovernewphenomena
thatwerenotpredictedbytheexistingtheory,andtoaccountforthese
wehavehadtodevelopamoreadvancedtheory.Itwouldthereforenot
beverysurprisingifthepresentgenerationofgrandunifiedtheorieswas
wronginclaimingthatnothingessentiallynewwillhappenbetweenthe
electroweak unification energy of about 100 GeV and the grand
unificationenergy of aboutathousand million million GeV.Wemight
indeedexpecttofindseveralnewlayersofstructuremorebasicthanthe
quarksandelectronsthatwenowregardas“elementary”particles.
However,itseemsthatgravitymayprovidealimittothissequenceof
“boxeswithinboxes.”Ifonehadaparticlewithanenergyabovewhatis
calledthePlanckenergy,tenmillionmillionmillionGeV(1followedby
nineteen zeros), its mass would be so concentrated that it would cut
itselfofffromtherestoftheuniverseandformalittleblackhole.Thus
itdoesseemthatthesequenceofmoreandmorerefinedtheoriesshould
have some limit as we go to higher and higher energies,so that there
should be some ultimate theory of the universe. Of course, the Planck
energyisaverylongwayfromtheenergiesofaroundahundredGeV,
whicharethemostthatwecanproduceinthelaboratoryatthepresent
time. We shall not bridge that gap with particle accelerators in the
foreseeablefuture!The veryearly stagesof theuniverse,however,are
anarenawheresuchenergiesmusthaveoccurred.Ithinkthatthereisa
goodchancethatthestudyoftheearlyuniverseandtherequirementsof
mathematical consistency will lead us to a complete unified theory
within the lifetime of some of us who are around today, always
presumingwedon’tblowourselvesupfirst.
Whatwoulditmeanifweactuallydiddiscovertheultimatetheoryof
theuniverse? Aswasexplained in Chapter1, we could never be quite
surethatwehadindeedfoundthecorrecttheory,sincetheoriescan’tbe
proved.Butifthetheorywasmathematicallyconsistentandalwaysgave
predictions that agreed with observations, we could be reasonably
confidentthatitwastherightone.Itwouldbringtoanendalongand
glorious chapter in the history of humanity’s intellectual struggle to
understand the universe. But it would also revolutionize the ordinary
person’sunderstandingofthelawsthatgoverntheuniverse.InNewton’s
timeitwaspossibleforaneducatedpersontohaveagraspofthewhole
ofhumanknowledge,atleastinoutline.Butsincethen,thepaceofthe
developmentofsciencehasmadethis impossible.Becausetheoriesare
alwaysbeingchanged toaccount fornew observations,they arenever
properlydigestedorsimplifiedsothat ordinarypeoplecan understand
them.Youhavetobeaspecialist,andeventhenyoucanonlyhopeto
have a proper grasp of a small proportion of the scientific theories.
Further,therateofprogressissorapidthatwhatonelearnsatschoolor
university is always a bit out of date. Only a few people can keep up
with the rapidly advancing frontier of knowledge, and they have to
devotetheirwholetimetoitandspecializeinasmallarea.Therestof
thepopulationhaslittleideaoftheadvancesthatarebeingmadeorthe
excitementtheyaregenerating.Seventyyearsago,ifEddingtonistobe
believed, only two people understood the general theory of relativity.
Nowadays tens of thousands of university graduates do, and many
millions of people are at least familiar with the idea. If a complete
unifiedtheorywasdiscovered,itwouldonlybeamatteroftimebefore
itwasdigestedandsimplifiedinthesamewayandtaughtinschools,at
leastinoutline.Wewouldthenallbeabletohavesomeunderstanding
of the laws that govern the universe and are responsible for our
existence.
Evenifwedodiscoveracompleteunifiedtheory,itwouldnotmean
thatwewouldbeabletopredicteventsingeneral,fortworeasons.The
first is the limitation that the uncertainty principle of quantum
mechanicssetsonourpowersofprediction.Thereisnothingwecando
to get around that. In practice, however, this first limitation is less
restrictivethanthesecondone.Itarisesfromthefactthatwecouldnot
solve the equations of the theory exactly, except in very simple
situations.(Wecannotevensolveexactlyforthemotionofthreebodies
in Newton’s theory of gravity, and the difficulty increases with the
numberofbodiesandthecomplexityofthetheory.)Wealreadyknow
the laws that govern the behavior of matter under all but the most
extremeconditions.Inparticular,weknowthebasiclawsthatunderlie
all of chemistry and biology. Yet we have certainly not reduced these
subjects to the status of solved problems: we have, as yet, had little
successinpredictinghumanbehaviorfrommathematicalequations!So
evenifwedofindacompletesetofbasiclaws,therewillstillbeinthe
years ahead the intellectually challenging task of developing better
approximationmethods,so thatwe canmakeuseful predictionsofthe
probableoutcomes in complicated and realistic situations. A complete,
consistent, unified theory is only the first step: our goal is a complete
understandingoftheeventsaroundus,andofourownexistence.
W
CHAPTER12
CONCLUSION
efindourselvesinabewilderingworld.Wewanttomakesenseof
what we see around us and to ask: What is the nature of the
universe?Whatisourplaceinitandwherediditandwecomefrom?
Whyisitthewayitis?
Totrytoanswerthesequestionsweadoptsome“worldpicture.”Just
as an infinite tower of tortoises supporting the flat earth is such a
picture, so is the theory of superstrings. Both are theories of the
universe,thoughthelatterismuchmoremathematicalandprecisethan
theformer.Boththeorieslackobservationalevidence:noonehasever
seenagianttortoisewiththeearthonitsback,butthen,noonehasseen
a superstring either. However, the tortoise theory fails to be a good
scientifictheorybecauseitpredictsthatpeopleshouldbeabletofalloff
theedgeoftheworld.Thishasnotbeenfoundtoagreewithexperience,
unless that turns out to be the explanation for the people who are
supposedtohavedisappearedintheBermudaTriangle!
Theearliesttheoreticalattemptstodescribeandexplaintheuniverse
involvedtheideathateventsandnaturalphenomenawerecontrolledby
spirits with human emotions who acted in a very humanlike and
unpredictablemanner.Thesespiritsinhabitednaturalobjects,likerivers
andmountains,includingcelestialbodies,likethesunandmoon.They
hadtobeplacatedandtheirfavorsoughtinordertoensurethefertility
ofthesoilandtherotationoftheseasons.Gradually,however,itmust
have been noticed that there were certain regularities: the sun always
roseintheeastandsetinthewest,whetherornotasacrificehadbeen
made to the sun god. Further, the sun, the moon, and the planets
followedprecisepathsacrosstheskythatcouldbepredictedinadvance
withconsiderableaccuracy.Thesunandthemoonmightstillbegods,
but they were gods who obeyed strict laws, apparently without any
exceptions, if one discounts stories like that of the sun stopping for
Joshua.
At first, these regularities and laws were obvious only in astronomy
and a few other situations. However, as civilization developed, and
particularlyinthelast300years,moreandmoreregularitiesandlaws
werediscovered.ThesuccessoftheselawsledLaplaceatthebeginning
ofthenineteenthcenturytopostulatescientificdeterminism;thatis,he
suggested that there would be a set of laws that would determine the
evolutionoftheuniverseprecisely,givenitsconfigurationatonetime.
Laplace’sdeterminismwasincompleteintwoways.Itdidnotsayhow
thelawsshouldbechosenanditdidnotspecifytheinitialconfiguration
of the universe. These were left to God. God would choose how the
universebeganandwhatlawsitobeyed,buthewouldnotintervenein
theuniverseonceithadstarted.Ineffect,Godwasconfinedtotheareas
thatnineteenth-centurysciencedidnotunderstand.
WenowknowthatLaplace’shopesofdeterminismcannotberealized,
at least in the terms he had in mind. The uncertainty principle of
quantummechanicsimpliesthatcertainpairsofquantities,suchasthe
position and velocity of a particle, cannot both be predicted with
complete accuracy. Quantum mechanics deals with this situation via a
class of quantum theories in which particles don’t have well-defined
positionsandvelocitiesbutarerepresentedbyawave.Thesequantum
theories are deterministic in the sense that they give laws for the
evolution of the wave with time. Thus if one knows the wave at one
time,onecancalculateitatanyothertime.Theunpredictable,random
elementcomesinonlywhenwetrytointerpretthewaveintermsofthe
positions and velocities of particles. But maybe that is our mistake:
maybetherearenoparticlepositionsandvelocities,butonlywaves.Itis
justthatwetrytofitthewavestoourpreconceivedideasofpositions
and velocities. The resulting mismatch is the cause of the apparent
unpredictability.
Ineffect,wehaveredefinedthetaskofsciencetobethediscoveryof
laws that will enable us to predict events up to the limits set by the
uncertaintyprinciple.Thequestionremains,however:howorwhywere
thelawsandtheinitialstateoftheuniversechosen?
InthisbookIhavegivenspecialprominencetothelawsthatgovern
gravity,becauseitisgravitythatshapesthelarge-scalestructureofthe
universe,eventhoughitistheweakestofthefourcategoriesofforces.
The laws of gravity were incompatible with the view held until quite
recentlythattheuniverseisunchangingintime:thefactthatgravityis
alwaysattractiveimpliesthattheuniversemustbeeitherexpandingor
contracting. According to the general theory of relativity, there must
have been a state of infinite density in the past, the big bang, which
wouldhavebeenaneffectivebeginningoftime.Similarly,ifthewhole
universe recollapsed, there must be another state of infinite density in
thefuture,thebigcrunch,whichwouldbeanendoftime.Evenifthe
whole universe did not recollapse, there would be singularities in any
localizedregionsthatcollapsedtoformblackholes.Thesesingularities
wouldbeanendoftimeforanyonewhofellintotheblackhole.Atthe
bigbangandothersingularities,allthelawswouldhavebrokendown,
soGodwouldstillhavehadcompletefreedomtochoosewhathappened
andhowtheuniversebegan.
Whenwe combine quantum mechanics with general relativity, there
seemstobeanewpossibilitythat didnotarisebefore:thatspaceand
time together might form a finite, four-dimensional space without
singularitiesorboundaries,likethesurfaceoftheearthbutwithmore
dimensions.Itseemsthatthisideacouldexplainmanyoftheobserved
featuresoftheuniverse,suchasitslarge-scaleuniformityandalsothe
smaller-scaledeparturesfromhomogeneity,likegalaxies,stars,andeven
human beings. It could even account for the arrow of time that we
observe. But if the universe is completely self-contained, with no
singularities or boundaries, and completely described by a unified
theory,thathasprofoundimplicationsfortheroleofGodasCreator.
Einsteinonceaskedthequestion:“HowmuchchoicedidGodhavein
constructing the universe?” If the no boundary proposal is correct, he
hadnofreedomatalltochooseinitialconditions.Hewould,ofcourse,
stillhavehadthefreedomtochoosethelawsthattheuniverseobeyed.
This,however,maynotreallyhavebeenallthatmuchofachoice;there
maywellbeonlyone,orasmallnumber,ofcompleteunifiedtheories,
suchastheheteroticstringtheory,thatareself-consistentandallowthe
existence of structures as complicated as human beings who can
investigatethelawsoftheuniverseandaskaboutthenatureofGod.
Evenifthereisonlyonepossibleunifiedtheory,itisjustasetofrules
andequations.Whatisitthatbreathesfireintotheequationsandmakes
a universe for them to describe? The usual approach of science of
constructingamathematicalmodelcannotanswerthequestionsofwhy
there should be a universe for the model to describe. Why does the
universe go to all the bother of existing? Is the unified theory so
compelling that it brings about its own existence? Or does it need a
creator,and,ifso,doeshehaveanyothereffectontheuniverse?And
whocreatedhim?
Up to now, most scientists have been too occupied with the
developmentofnewtheoriesthatdescribewhattheuniverseistoaskthe
questionwhy.Ontheotherhand,thepeoplewhosebusinessitistoask
why,thephilosophers,havenotbeenabletokeepupwiththeadvance
ofscientifictheories.Intheeighteenthcentury,philosophersconsidered
thewholeofhumanknowledge,includingscience,tobetheirfieldand
discussed questions such as: did the universe have a beginning?
However,inthenineteenthandtwentiethcenturies,sciencebecametoo
technicalandmathematicalforthephilosophers,oranyoneelseexcepta
few specialists. Philosophers reduced the scope of their inquiries so
much that Wittgenstein, the most famous philosopher of this century,
said, “The sole remaining task for philosophy is the analysis of
language.” What a comedown from the great tradition of philosophy
fromAristotletoKant!
However, if we do discover a complete theory, it should in time be
understandableinbroadprinciplebyeveryone,notjustafewscientists.
Thenweshallall,philosophers,scientists,andjustordinarypeople,be
abletotakepartinthediscussionofthequestionofwhyitisthatwe
and the universe exist. If we find the answer to that, it would be the
ultimatetriumphofhumanreason—forthenwewouldknowthemind
ofGod.
E
ALBERTEINSTEIN
instein’s connection with the politics of the nuclear bomb is well
known:he signedthefamous letter toPresident Franklin Roosevelt
that persuaded the United States to take the idea seriously, and he
engagedin postwar efforts to preventnuclearwar. But these were not
justtheisolatedactionsofascientistdraggedintotheworldofpolitics.
Einstein’s life was, in fact, to use his own words, “divided between
politicsandequations.”
Einstein’searliestpoliticalactivitycameduringtheFirstWorldWar,
whenhewasaprofessorinBerlin.Sickenedbywhathesawasthewaste
of human lives, he became involved in antiwar demonstrations. His
advocacy of civil disobedience and public encouragement of people to
refuse conscription did little to endear him to his colleagues. Then,
following the war, he directed his efforts toward reconciliation and
improving international relations. This too did not make him popular,
andsoonhispoliticsweremakingitdifficultforhimtovisittheUnited
States,eventogivelectures.
Einstein’ssecondgreatcausewasZionism.AlthoughhewasJewishby
descent,EinsteinrejectedthebiblicalideaofGod.However,agrowing
awarenessofanti-Semitism,bothbeforeandduringtheFirstWorldWar,
ledhimgraduallytoidentifywith theJewishcommunity,andlaterto
becomeanoutspokensupporterofZionism.Oncemoreunpopularitydid
notstophimfromspeakinghismind.Histheoriescameunderattack;an
anti-Einstein organization was even set up. One man was convicted of
inciting others to murder Einstein (and fined a mere six dollars). But
Einstein was phlegmatic. When a book was published entitled 100
AuthorsAgainstEinstein,he retorted, “IfI were wrong,thenone would
havebeenenough!”
In1933,Hitlercametopower.EinsteinwasinAmerica,anddeclared
he would not return to Germany. Then, while Nazi militia raided his
house and confiscated his bank account, a Berlin newspaper displayed
theheadline“GoodNewsfromEinstein—He’sNotComingBack.”Inthe
face of the Nazi threat, Einstein renounced pacifism, and eventually,
fearing that German scientists would build a nuclear bomb, proposed
thattheUnitedStatesshoulddevelopitsown.Butevenbeforethefirst
atomic bomb had been detonated, he was publicly warning of the
dangers of nuclear war and proposing international control of nuclear
weaponry.
Throughouthislife,Einstein’seffortstowardpeaceprobablyachieved
little that would last—and certainly won him few friends. His vocal
support of the Zionist cause, however, was duly recognized in 1952,
when he was offered the presidency of Israel. He declined, saying he
thought he was too naive in politics. But perhaps his real reason was
different: to quote him again, “Equations are more important to me,
because politics is for the present, but an equation is something for
eternity.”
G
GALILEOGALILEI
alileo,perhapsmore thananyother singleperson,wasresponsible
for the birth of modern science. His renowned conflict with the
CatholicChurchwascentraltohisphilosophy,forGalileowasoneofthe
firsttoarguethatmancouldhopetounderstandhowtheworldworks,
and,moreover,thatwecoulddothisbyobservingtherealworld.
Galileohadbelieved Copernicantheory(that theplanetsorbitedthe
sun)sinceearlyon,butitwasonlywhenhefoundtheevidenceneeded
tosupporttheideathathestartedtopubliclysupportit.Hewroteabout
Copernicus’stheoryinItalian(nottheusualacademicLatin),andsoon
his views became widely supported outside the universities. This
annoyedtheAristotelianprofessors,whounitedagainsthimseekingto
persuadetheCatholicChurchtobanCopernicanism.
Galileo,worriedbythis,traveledtoRometospeaktotheecclesiastical
authorities.HearguedthattheBiblewasnotintendedtotellusanything
aboutscientifictheories,andthatitwasusualtoassumethat,wherethe
Bible conflicted with common sense, it was being allegorical. But the
Church was afraid of a scandal that might undermine its fight against
Protestantism, and so took repressive measures. It declared
Copernicanism“falseanderroneous” in1616, andcommanded Galileo
neveragainto“defendorhold”thedoctrine.Galileoacquiesced.
In1623,alongtimefriendofGalileo’sbecamethePope.Immediately
Galileo tried to get the 1616 decree revoked. He failed, but he did
manage to get permission to write a book discussing both Aristotelian
andCopernicantheories,ontwoconditions:hewouldnottakesidesand
wouldcometotheconclusionthatmancouldinanycasenotdetermine
howtheworldworkedbecauseGodcouldbringaboutthesameeffects
inwaysunimaginedbyman,whocouldnotplacerestrictionsonGod’s
omnipotence.
The book, Dialogue Concerning the Two Chief World Systems, was
completedandpublishedin1632,withthefullbackingofthecensors—
and was immediately greeted throughout Europe as a literary and
philosophical masterpiece. Soon the Pope, realizing that people were
seeing the book as a convincing argument in favor of Copernicanism,
regrettedhavingalloweditspublication.ThePopearguedthatalthough
thebookhadtheofficialblessingofthecensors,Galileohadnevertheless
contravenedthe1616decree.HebroughtGalileobeforetheInquisition,
who sentenced him to house arrest for life and commanded him to
publiclyrenounceCopernicanism.Forasecondtime,Galileoacquiesced.
GalileoremainedafaithfulCatholic,buthisbeliefintheindependence
of science had not been crushed. Four years before his death in 1642,
whilehewasstillunderhousearrest,themanuscriptofhissecondmajor
bookwassmuggledtoapublisherinHolland.Itwasthiswork,referred
toasTwoNewSciences,evenmorethanhissupportforCopernicus,that
wastobethegenesisofmodernphysics.
I
ISAACNEWTON
saac Newton was not a pleasant man. His relations with other
academicswerenotorious,withmostofhislaterlifespentembroiled
in heated disputes. Following publication of Principia Mathematica—
surely the most influential book ever written in physics—Newton had
risenrapidlyintopublicprominence.Hewasappointedpresidentofthe
RoyalSocietyandbecamethefirstscientistevertobeknighted.
Newton soon clashed with the Astronomer Royal, John Flamsteed,
whohadearlierprovidedNewtonwithmuch-neededdataforPrincipia,
but was now withholding information that Newton wanted. Newton
would not take no for an answer: he had himself appointed to the
governing body of the Royal Observatory and then tried to force
immediate publication of the data. Eventually he arranged for
Flamsteed’s work to be seized and prepared for publication by
Flamsteed’smortalenemy,EdmondHalley.ButFlamsteedtookthecase
to court and, in the nick of time, won a court order preventing
distribution of the stolen work. Newton was incensed and sought his
revenge by systematically deleting all references to Flamsteed in later
editionsofPrincipia.
AmoreseriousdisputearosewiththeGermanphilosopherGottfried
Leibniz. Both Leibniz and Newton had independently developed a
branchofmathematicscalledcalculus,whichunderliesmostofmodern
physics.AlthoughwenowknowthatNewtondiscoveredcalculusyears
beforeLeibniz,hepublishedhisworkmuchlater.Amajorrowensued
over who had been first, with scientists vigorously defending both
contenders. It is remarkable, however, that most of the articles
appearingindefenseofNewtonwereoriginallywrittenbyhisownhand
—andonlypublishedinthenameoffriends!Astherowgrew,Leibniz
made the mistake of appealing to the Royal Society to resolve the
dispute. Newton, as president, appointed an “impartial” committee to
investigate, coincidentally consisting entirely of Newton’s friends! But
thatwasnotall:Newtonthenwrotethecommittee’sreporthimselfand
had the Royal Society publish it, officially accusing Leibniz of
plagiarism.Stillunsatisfied,hethenwroteananonymousreviewofthe
report in the Royal Society’s own periodical. Following the death of
Leibniz, Newton is reported to have declared that he had taken great
satisfactionin“breakingLeibniz’sheart.”
During the period of these two disputes, Newton had already left
Cambridgeandacademe.Hehadbeenactiveinanti-Catholicpoliticsat
Cambridge,andlaterinParliament,andwasrewardedeventuallywith
thelucrativepostofWardenoftheRoyalMint.Hereheusedhistalents
for deviousness and vitriol in a more socially acceptable way,
successfully conducting a major campaign against counterfeiting, even
sendingseveralmentotheirdeathonthegallows.
GLOSSARY
Absolute zero: The lowest possible temperature, at which substances
containnoheatenergy.
Acceleration:Therateatwhichthespeedofanobjectischanging.
Anthropic principle: We see the universe the way it is because if it
weredifferentwewouldnotbeheretoobserveit.
Antiparticle: Each type of matter particle has a corresponding
antiparticle. When a particle collides with its antiparticle, they
annihilate,leavingonlyenergy.
Atom: The basic unit of ordinary matter, made up of a tiny nucleus
(consistingofprotonsandneutrons)surroundedbyorbitingelectrons.
Bigbang:Thesingularityatthebeginningoftheuniverse.
Bigcrunch:Thesingularityattheendoftheuniverse.
Blackhole:Aregionofspace-timefromwhichnothing,notevenlight,
canescape,becausegravityissostrong.
Casimireffect:Theattractivepressurebetweentwoflat,parallelmetal
platesplacedveryneartoeachotherinavacuum.Thepressureisdueto
areductionintheusualnumberofvirtualparticlesinthespacebetween
theplates.
Chandrasekharlimit:Themaximumpossiblemassofastablecoldstar,
abovewhichitmustcollapseintoablackhole.
Conservationofenergy:Thelawofsciencethatstatesthatenergy(or
itsequivalentinmass)canneitherbecreatednordestroyed.
Coordinates:Numbersthatspecifythepositionofapointinspaceand
time.
Cosmologicalconstant:AmathematicaldeviceusedbyEinsteintogive
space-timeaninbuilttendencytoexpand.
Cosmology:Thestudyoftheuniverseasawhole.
Darkmatter:Matteringalaxies,clusters,andpossiblybetweenclusters,
thatcannotbeobserveddirectlybutcanbedetectedbyitsgravitational
effect.Asmuchas90percentofthemassoftheuniversemaybeinthe
formofdarkmatter.
Duality: A correspondence between apparently different theories that
leadtothesamephysicalresults.
Einstein-Rosen bridge: A thin tube of space-time linking two black
holes.AlsoseeWormhole.
Electric charge: A property of a particle by which it may repel (or
attract)otherparticlesthathaveachargeofsimilar(oropposite)sign.
Electromagnetic force: The force that arises between particles with
electriccharge;thesecondstrongestofthefourfundamentalforces.
Electron:Aparticlewithnegativeelectricchargethatorbitsthenucleus
ofanatom.
Electroweakunificationenergy:Theenergy(around100GeV)above
which the distinction between the electromagnetic force and the weak
forcedisappears.
Elementary particle: A particle that, it is believed, cannot be
subdivided.
Event:Apointinspace-time,specifiedbyitstimeandplace.
Eventhorizon:Theboundaryofablackhole.
Exclusionprinciple:Theideathattwoidenticalspin-½particlescannot
have(withinthelimitssetbytheuncertaintyprinciple)boththesame
positionandthesamevelocity.
Field:Somethingthatexiststhroughoutspaceandtime,asopposedtoa
particlethatexistsatonlyonepointatatime.
Frequency:Forawave,thenumberofcompletecyclespersecond.
Gammarays:Electromagneticraysofveryshortwavelength,produced
inradioactivedecayorbycollisionsofelementaryparticles.
Generalrelativity:Einstein’stheorybasedontheideathatthelawsof
science should be the same for all observers, no matter how they are
moving. It explains the force of gravity in terms of the curvature of a
four-dimensionalspace-time.
Geodesic:Theshortest(orlongest)pathbetweentwopoints.
Grandunificationenergy:Theenergyabovewhich,itisbelieved,the
electromagnetic force, weak force, and strong force become
indistinguishablefromeachother.
Grand unified theory (GUT): A theory which unifies the
electromagnetic,strong,andweakforces.
Imaginarytime:Timemeasuredusingimaginarynumbers.
Light cone: A surface in space-time that marks out the possible
directionsforlightrayspassingthroughagivenevent.
Light-second(light-year):Thedistancetraveledbylightinonesecond
(year).
Magnetic field: The field responsible for magnetic forces, now
incorporatedalongwiththeelectricfield,intotheelectromagneticfield.
Mass: The quantity of matter in a body; its inertia, or resistance to
acceleration.
Microwavebackgroundradiation:Theradiationfromtheglowingof
thehotearlyuniverse,nowsogreatlyred-shiftedthatitappearsnotas
light but as microwaves (radio waves with a wavelength of a few
centimeters).AlsoseeCOBE,onthispage.
Nakedsingularity:Aspace-timesingularitynotsurroundedbyablack
hole.
Neutrino:Anextremelylight(possiblymassless)particlethatisaffected
onlybytheweakforceandgravity.
Neutron: An uncharged particle, very similar to the proton, which
accountsforroughlyhalftheparticlesinanatomicnucleus.
Neutron star: A cold star, supported by the exclusion principle
repulsionbetweenneutrons.
Noboundarycondition:Theideathattheuniverseisfinitebuthasno
boundary(inimaginarytime).
Nuclearfusion:Theprocessbywhichtwonucleicollideandcoalesceto
formasingle,heaviernucleus.
Nucleus: The central part of an atom, consisting only of protons and
neutrons,heldtogetherbythestrongforce.
Particle accelerator: A machine that, using electromagnets, can
acceleratemovingchargedparticles,givingthemmoreenergy.
Phase: For a wave, the position in its cycle at a specified time: a
measureofwhetheritisatacrest,atrough,orsomewhereinbetween.
Photon:Aquantumoflight.
Planck’squantumprinciple:Theideathatlight(oranyotherclassical
waves) can be emitted or absorbed only in discrete quanta, whose
energyisproportionaltotheirwavelength.
Positron:The(positivelycharged)antiparticleoftheelectron.
Primordialblackhole:Ablackholecreatedintheveryearlyuniverse.
Proportional:‘XisproportionaltoY’meansthatwhenYismultiplied
by any number, so is X. ‘X is inversely proportional to Y’ means that
whenYismultipliedbyanynumber,Xisdividedbythatnumber.
Proton:Apositivelychargedparticle,verysimilartotheneutron,that
accountsforroughlyhalftheparticlesinthenucleusofmostatoms.
Pulsar:Arotatingneutronstarthatemitsregularpulsesofradiowaves.
Quantum: The indivisible unit in which waves may be emitted or
absorbed.
Quantum chromodynamics (QCD): The theory that describes the
interactionsofquarksandgluons.
Quantum mechanics: The theory developed from Planck’s quantum
principleandHeisenberg’suncertaintyprinciple.
Quark: A (charged) elementary particle that feels the strong force.
Protonsandneutronsareeachcomposedofthreequarks.
Radar: A system using pulsed radio waves to detect the position of
objectsbymeasuringthetimeittakesasinglepulsetoreachtheobject
andbereflectedback.
Radioactivity: The spontaneous breakdown of one type of atomic
nucleusintoanother.
Redshift:Thereddeningoflightfromastarthatismovingawayfrom
us,duetotheDopplereffect.
Singularity: A point in space-time at which the space-time curvature
becomesinfinite.
Singularitytheorem:Atheoremthatshowsthatasingularitymustexist
undercertaincircumstances—inparticular,thattheuniversemusthave
startedwithasingularity.
Space-time:Thefour-dimensionalspacewhosepointsareevents.
Spatialdimension:Anyofthethreedimensionsthatarespacelike—that
is,anyexceptthetimedimension.
Specialrelativity:Einstein’stheorybasedontheideathatthelawsof
science should be the same for all observers, no matter how they are
moving,intheabsenceofgravitationalphenomena.
Spectrum:Thecomponentfrequenciesthatmakeupawave.Thevisible
partofthesun’sspectrumcanbeseeninarainbow.
Spin: An internal property of elementary particles, related to, but not
identicalto,theeverydayconceptofspin.
Stationarystate:Onethatisnotchangingwithtime:aspherespinning
at a constant rate is stationary because it looks identical at any given
instant.
Stringtheory: A theory of physics in which particles are described as
wavesonstrings.Stringshavelengthbutnootherdimension.
Strong force: The strongest of the four fundamental forces, with the
shortest range of all. It holds the quarks together within protons and
neutrons,andholdstheprotonsandneutronstogethertoformatoms.
Uncertainty principle: The principle, formulated by Heisenberg, that
onecanneverbeexactlysureofboththepositionandthevelocityofa
particle;themoreaccuratelyoneknowstheone,thelessaccuratelyone
canknowtheother.
Virtual particle: In quantum mechanics, a particle that can never be
directlydetected,butwhoseexistencedoeshavemeasurableeffects.
Wave/particleduality:Theconceptinquantummechanicsthatthereis
no distinction between waves and particles; particles may sometimes
behavelikewaves,andwaveslikeparticles.
Wavelength:Forawave,thedistancebetweentwoadjacenttroughsor
twoadjacentcrests.
Weakforce:Thesecondweakestofthefourfundamentalforces,witha
very short range. It affects all matter particles, but not force-carrying
particles.
Weight: The force exerted on a body by a gravitational field. It is
proportionalto,butnotthesameas,itsmass.
White dwarf: A stable cold star, supported by the exclusion principle
repulsionbetweenelectrons.
Wormhole:Athintubeofspace-timeconnectingdistantregionsofthe
universe. Wormholes might also link to parallel or baby universes and
couldprovidethepossibilityoftimetravel.
ACKNOWLEDGMENTS
Many people have helped me in writing this book. My scientific
colleagues have without exception been inspiring. Over the years my
principal associates and collaborators were Roger Penrose, Robert
Geroch,BrandonCarter,GeorgeEllis,GaryGibbons,DonPage,andJim
Hartle. I owe a lot to them, and to my research students, who have
alwaysgivenmehelpwhenneeded.
Oneofmystudents,BrianWhitt,gavemealotofhelpwritingthefirst
editionofthisbook.MyeditoratBantamBooks,PeterGuzzardi,made
innumerable comments which improved the book considerably. In
addition, for this edition, I would like to thank Andrew Dunn, who
helpedmerevisethetext.
Icouldnothavewrittenthisbookwithoutmycommunicationsystem.
Thesoftware,calledEqualizer,wasdonatedbyWaltWaltoszofWords
PlusInc.,inLancaster,California.Myspeechsynthesizerwasdonatedby
Speech Plus, of Sunnyvale, California. The synthesizer and laptop
computer were mounted on my wheelchair by David Mason, of
Cambridge Adaptive Communication Ltd. With this system I can
communicatebetternowthanbeforeIlostmyvoice.
I have had a number of secretaries and assistants over the years in
which I wrote and revised this book. On the secretarial side, I’m very
gratefultoJudyFella,AnnRalph,LauraGentry,CherylBillington,and
SueMasey.MyassistantshavebeenColinWilliams,DavidThomas,and
Raymond Laflamme, Nick Phillips, Andrew Dunn, Stuart Jamieson,
Jonathan Brenchley, Tim Hunt, Simon Gill, Jon Rogers, and Tom
Kendall.They,mynurses,colleagues,friends,andfamilyhaveenabled
me to live a very full life and to pursue my research despite my
disability.
StephenHawking
ABOUTTHEAUTHOR
STEPHENHAWKINGwastheLucasianProfessorofMathematicsatthe
UniversityofCambridgeforthirtyyears,andhasbeentherecipientof
numerousawardsandhonorsincluding,mostrecently,thePresidential
MedalofFreedom.HisbooksforthegeneralreaderincludetheclassicA
BriefHistoryofTime,theessaycollectionBlackHolesandBabyUniverses,
TheUniverseinaNutshell,ABrieferHistoryofTimeandTheGrandDesign.